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is  a  revised  translation  by  Ivo  Thomas  of  the  German  edition,  Formale  Logik, 

by  J.  M.  Bochehski,  published  and  copyrighted  by  Verlag  Karl  Alber, 

Freiburg/Miinchen,  in  1956 

©   1961  by 

University  of  Notre  Dame  Press,  Notre  Dame,  Indiana 

Library  of  Congress  Catalog  Card  Number  58 — 14183 

Printed  in  the  U.  S.A. 


This  history  of  the  problems  of  formal  logic,  which  we  believe 
to  be  the  first  comprehensive  one,  has  grown  only  in  small  part  from 
the  author's  own  researches.  Its  writing  has  been  made  possible 
by  a  small  group  of  logicians  and  historians  of  logic,  those  above 
all  of  the  schools  of  Warsaw  and  Munster.  It  is  the  result  of  their 
labours  that  the  work  chiefly  presents,  and  the  author  offers  them 
his  thanks,  especially  to  the  founders  Jan  Lukasiewicz  and  Heinrich 

A  whole  series  of  scholars  has  been  exceptionally  obliging  in 
giving  help  with  the  compilation.  Professors  E.  W.  Beth  (Amster- 
dam), Ph.  Boehner  O.F.M.  (St.  Bonaventure,  N.Y.),  A.  Church 
(Princeton),  0.  Gigon  (Bern),  D.  Ingalls  (Harvard),  J.  Lukasiewicz 
(Dublin),  B.  Mates  (Berkeley,  California),  E.  Moody  (Columbia 
University,  New  York),  M.  Morard  O.P.  (Fribourg),  C.  Regamey 
(Fribourg/Lausanne)  and  I.  Thomas  O.P.  (Blackfriars,  Oxford)  have 
been  kind  enough  to  read  various  parts  of  the  manuscript  and 
communicate  to  me  many  valuable  remarks,  corrections  and  addi- 
tions. Thanks  to  them  I  have  been  able  to  remove  various  inexacti- 
tudes and  significantly  improve  the  content.  Of  course  they  bear  no 
responsibility  for  the  text  in  its  final  state. 

The  author  is  further  indebted  for  important  references  and 
information  to  Mile.  M.  T.  d'Alverny,  Reader  of  the  Department  of 
Manuscripts  of  the  Bibliotheque  Nationale  in  Paris,  Dr.  J.  Yajda 
of  the  Centre  National  de  la  Recherche  Scientifique  in  Paris,  Pro- 
fessors L.  Minio-Paluello  (Oxford),  S.  Hulsewe  (Leiden),  H.  Hermes 
and  H.  Scholz  (Munster  i.  W.),  R.  Feys  (Louvain)  and  A.  Badawi 
(Fuad  University,  Cairo).  Dr.  A.  Menne  has  been  kind  enough  to 
read  the  proofs  and  make  a  number  of  suggestions. 

My  assistant,  Dr.  Thomas  Raber,  has  proved  a  real  collaborator 
throughout.  In  particular,  I  could  probably  not  have  achieved  the 
translation  of  the  texts  into  German  without  his  help.  He  has 
also  been  especially  helpful  in  the  compilation  of  the  Bibliography 
and  the  preparation  of  the  manuscript  for  press. 


In  the  course  of  my  researches  I  have  enjoyed  the  help  of  several 
European  libraries.  I  should  like  to  name  here  those  in  Amsterdam 
(University  Library),  Basel  (University  Library),  Gottingen  (Nie- 
dersachsische  Landes-  und  Universitatsbibliothek),  Kolmar  (Stadt- 
bibliothek),  London  (British  Museum  and  India  Office  Library), 
Munich  (Bayerische  Staatsbibliothek),  Oxford  (Bodleian  Library) 
and  Paris  (Bibliotheque  Nationale);  above  all  the  Kern-Institut  in 
Leiden  and  the  institutes  for  mathematical  logic  in  Louvain  and 
Munster  which  showed  me  notable  hospitality.  Finally,  last  but 
not  least,  the  Cantonal  and  University  Library  of  Fribourg,  where 
the  staff  has  made  really  extraordinary  efforts  on  my  behalf. 

The  completion  of  my  inquiries  and  the  composition  of  this  book 
was  made  materially  possible  by  a  generous  grant  from  the  Swiss 
national  fund  for  the  advancement  of  scientific  research.  This 
enabled  me  to  employ  an  assistant  and  defray  the  costs  of  several 
journeys,  of  microfilms  etc.  My  best  thanks  are  due  to  the  adminis- 
trators of  the  fund,  as  to  all  who  have  helped  me  in  the  work. 

Since  the  manuscript  was  completed,  Fr.  Ph.  Boehner  O.F.M. 
and  Dr  Richard  Brodfiihrer,  editor  of  the  series  'Orbis  Academicus', 
have  died.  Both  are  to  be  remembered  with  gratitude. 





In  this  edition  of  the  most  considerable  history  of  formal  logic 
yet  published,  the  opportunity  has  of  course  been  taken  to  make 
some  adjustments  seen  to  be  necessary  in  the  original,  with  the 
author's  full  concurrence.  Only  in  §  36,  however,  has  the  numeration 
of  cited  passages  been  altered  owing  to  the  introduction  of  new 
matter.  Those  changes  are  as  follows: 

German  edition  English  edition 

36.13  36.17 

36.14  36.18 
36.15—17                                36.21—23 

Other  alterations  that  may  be  noted  are:  the  closing  paragraphs  of 
§  15  have  been  more  accurately  suited  to  the  group  of  syllogisms 
which  they  concern;  16.17  appears  here  as  a  principle  rejected,  not 
accepted,  by  Aristotle;  27.28  has  been  re-interpreted  and  the 
lengthy  citation  dropped;  a  new  sub-section,  on  the  beginnings  of 
combinatory  logic,  has  been  added  to  §  49.  A  few  further  items  have 
been  added  to  the  Bibliography.  On  grounds  of  economy  this  last 
has  been  reproduced  photographically;  probably  such  German 
remarks  and  abbreviations  as  it  contains  will  not  much  inconven- 
ience its  users. 

A  word  needs  to  be  said  about  §  5  A,  'Technical  Expressions', 
which  has  naturally  had  to  be  largely  re-written.  In  the  orginal. 
the  author  expressed  his  intention  of  using  'Aussage'  as  a  name  for 
sentential  expressions,  and  so  for  certain  dispositions  of  black  ink, 
or  bundles  of  sound-waves.  So  understood,  the  word  could  be 
treated  as  generally  synonymous  writh  the  Scholastic  'proposilio' 
and  the  English  'sentence'  when  used  in  an  equally  technical  sense, 
and  was  deemed  a  tolerable  translation  of  Russell's  'proposition'  the 
reference  of  which  is  often  ambiguous.  But  these  equations  evidently 
cannot  be  maintained  here;  for  one  thing,  they  would  warrant  the 
change  of  'proposition'  to   'sentence'  throughout  quotations  from 



Russell;  secondly  we  prefer,  with  A.  Church  (1.01),  to  speak  of 
'propositional  logic'  rather  than  'sentential  logic' ;  and  thirdly, 
one  risks  actual  falsification  of  one's  material  by  imposing  on  it  a 
grid  of  sharp  distinctions  which  for  the  most  part  belongs  to  a 
later  period  than  anything  here  treated.  As  noted  in  the  body  of 
the  work,  the  Stoics  and  Frege  were  alone  in  making  the  distinction 
between  sign  and  significate  as  sharply  as  is  now  customary.  So  we 
have  normally  used  'proposition'  where  the  author  used  'Aussage'  - 
always,  indeed,  when  the  Scholastic  'propositio'  needs  translation. 
In  Part  V  usage  is  of  course  largely  conditioned  by  the  fact  that  so 
many  citations  now  appear  in  their  original  language. 


As  to  the  contents,  an  evident  lacuna  is  the  absence  of  any  texts 
from  the  12th  century  a.d.,  and  the  author  himself  has  suggested 
that  30.03  is  quite  insufficient  reference  to  Peter  Abelard,  described 
in  an  epitaph  as  'the  Aristotle  of  our  time,  the  equal  or  superior  of 
all  logicians  there  have  been'  (1.02),  and  in  similar  words  by  John 
of  Salisbury  (1.03).  We  propose  only  to  elaborate  that  single  reference, 
by  way  of  giving  a  taste  of  this  twelfth  century  logic,  closely  based 
on  Boethius  in  its  past,  growing  in  an  atmosphere  of  keen  discus- 
sion in  its  present,  evidently  holding  the  seeds  of  later  Scholastic 
developments  as  exemplified  in  this  book. 

Abelard's  consequences  are  certainly  not  fully  emancipated  from 
the  logic  of  terms  (cf.  30.03)  yet  he  realizes  that  propositions  are 
always  involved.  His  explanation  of  'consequential'  may  first  be 
noted : 

1.04  A  hypothetical  proposition  is  called  a  'consequence' 
after  its  consequent,  and  a  'conditional'  after  its  condition. 

Speaking  later  of  a  form  of  the  laws  of  transposition  (43.33)  he 

1.05  My  opinion  is  that  while  the  force  of  the  inference  lies 
in  the  terms,  yet  the  whole  proposition  is  to  be  denied.  .  .  . 
Rightly  the  whole  sequent  and  antecedent  proposition  is  to 
be  denied,  since  the  inference  lies  between  the  entire  proposi- 
tions, though  the  force  of  the  inference  depends  on  the 
terms.  ...  So  that  the  hypothetical  proposition  is  rightly 
said  not  to  be  composed  of  simple  terms,  but  to  be  conjoined 
from  several  propositions,  inasmuch  as  it  propounds  that  what 
the  sequent  proposition  manifests,  follows  from  what  the 
precedent  (manifests).  So  that  the  denial  is  not  to  be  effected 
according  to  the  terms  alone,  but  according  to  the  entire 
propositions  between  which  the  relation  of  consequence  is 



Consequences  themselves  are  distinguished  from  their  metalogical 
formulations  (cf.  the  commentary  preceding  31.14),  the  latter  being 

called  'maximae  propositiones'  and  defined  thus: 

1.06  That  proposition  which  contains  the  sense  of  many 
consequences  and  manifests  the  manner  of  proof  common  to 
their  determining  features  (differentiae)  according  to  the 
force  of  their  relationship,  is  called  a  'maximal  proposition1. 
E.g.  along  with  these  consequences:  'if  it  is  man,  it  is  animal', 
'if  it  is  pearl,  it  is  stone',  'if  it  is  rose,  it  is  flower',  'if  it  is 
redness,  it  is  colour'  etc.,  in  which  species  precede  genera, 
a  maximal  proposition  such  as  the  following  is  adduced :  of 
whatever  the  species  is  predicated,  the  genus  too  (is  predi- 
cated). .  .  .  This  maximal  proposition  contains  and  expresses 
the  sense  of  all  such  consequences  and  manifests  the  way  of 
yielding  inference  common  to  the  antecedents. 

There  is  a  rich  store  of  maxims  in  Abelard,  but  it  is  not  always 
easy  to  see  whether  they  belong  to  the  logic  of  terms  or  propositions. 
This  ambiguity  has  been  noted  with  reference  to  Kilwardby  (cf. 
§  31 ,  B )  where  one  might  be  tempted  to  think  that  it  was  unconscious. 
But  the  terminology  is  not  subject  to  direct  attention  in  Kilwardby; 
in  Abelard  it  is,  and  the  ambiguity  is  noted  and  accepted.  The 
following  passage  may  need  apology  for  its  length,  but  not  for  its 
great  interest  in  respect  of  terminology,  semantic  considerations 
(on  which  we  cannot  here  delay),  maxims  both  of  validity  and 
invalidity,  and  the  reduction  of  some  of  them  to  others,  1.06  and 
1.07  are  enough  to  establish  the  basis  and  essentials  of  §  31  firmly 
in  the  12th  century. 

1.07  'Antecedent'  and  'consequent'  are  sometimes  used  to 
designate  complete  enunciations  as  when  in  the  consequence : 
if  Socrates  is  man,  Socrates  is  animal,  we  say  that  the  first 
categorical  is  antecedent  to  the  second;  sometimes  in  the 
designation  of  simple  terms  (dictio)  or  what  they  signify,  as 
when  we  say  in  regard  to  the  same  consequence  that  the 
species  is  antecedent  to  the  genus,  i.e.  'man'  to  'animal',  the 
nature  or  relationship  provides  inferential  force.  .  .  .  But 
whether  we  take  'antecedent'  and  'consequent'  for  simple 
terms  or  complete  enunciations,  we  can  call  them  the  parts  of 
hypothetical  enunciations,  i.e.  the  parts  of  which  the  conse- 
quences are  composed  and  of  which  they  consist,  not  parts 
of  which  they  treat.  For  we  cannot  accept  as  true  this  conse- 
quence: if  he  is  man,  he  is  animal,  if  it  treats  of  utterances 
(vocibus)  be  they  terms  or  propositions.  For  it  is  false  that  if 



the  utterance  'man'  exists,  there  should  also  be  the  utterance 
'animal';  and  similarly  in  the  case  of  enunciations  or  their 
concepts  (intelledibus).  For  it  is  not  necessary  that  he  who  has 
a  concept  generated  by  the  precedent  proposition  should  also 
have  one  generated  from  the  consequent.  For  no  diverse  con- 
cepts are  so  akin  that  one  must  be  possessed  along  with  the 
other;  indeed  everyone's  own  experience  will  convince  him 
that  his  soul  does  not  retain  diverse  concepts  and  will  find 
that  it  is  totally  occupied  with  each  single  concept  while  he 
has  it.  But  if  someone  were  to  grant  that  the  essences  of 
concepts  follow  on  one  another  like  the  essences  of  the 
things  from  which  the  concepts  are  gained,  he  would  have 
to  concede  that  every  knower  has  an  infinity  of  concepts 
since  every  proposition  has  innumerable  consequences. 
Further,  whether  we  treat  of  enunciations  or  of  their  concepts, 
we  have  to  use  their  names  in  a  consequence;  but  if  'man'  or 
'animal'  are  taken  as  names  either  of  enunciations  or  concepts, 
'if  there  is  man  there  is  animal'  cannot  at  all  be  a  consequence, 
being  composed  entirely  of  terms,  as  much  as  to  say:  'if  man 
animal' ;  indeed  as  a  statement  it  is  quite  imperfect.  To  keep, 
therefore,  a  genuine  relation  of  consequence  we  must  concede 
that  it  is  things  which  are  being  treated  of,  and  accept  the 
rules  of  antecedent  and  consequent  as  given  in  the  nature 
of  things.  These  rules  are  as  follows  : 

(1)  on  the  antecedent  being  posited,  the  consequent  is  posited; 

(2)  on  the  consequent  being  destroyed,  the  antecedent  is 
destroyed,  thus: 

'if  there  is  man  there  is  animal',  'if  there  is  not  animal  there 
is  not  man'; 

(3)  neither  if  the  antecedent  is  posited,  is  the  consequent 

(4)  nor  if  the  antecedent  is  destroyed  need  the  consequent 
be  destroyed 

(5)  or  posited,  just  as 

(6)  neither  if  the  consequent  is  destroyed  is  the  antecedent 

(7)  nor  if  the  same  (the  consequent)  is  posited  is  it  (the  ante- 
cedent) either  posited 

(8)  or  removed. 

Since  the  last  ( (6)-(8) )  are  equivalent  to  the  former  ( (3)-(5) ) 
as  also  their  affirmatives  are  mutually  equivalent,  the  two 
sets  must  be  simultaneously  true  or  false.  The  two  first  rules 


are  also  in  complete  mutual  agreement  and  can  be  derived 
from  one  another,  e.g.  if  it  is  conceded :  if  there  is  man  there 
is  animal,  it  must  also  be  conceded:  'if  there  is  not  animal 
there  is  not  man,  and  conversely. 

When  the  first  is  true,  the  second  will  be  proved  true  as 
follows,  by  inducing  an  impossibility.  Let  us  posit  this  as 
true:  if  there  is  man  there  is  animal,  and  doubt  about  this: 
if  there  is  not  animal  there  is  not  man,  i.e.  whether  'animal' 
negated  negates  'man'.  We  shall  confirm  this  in  the  following 
way.  Either  'animal'  negated  negates  'man'  or  negated  it 
admits  'man',  so  that  it  may*  happen  that  when  'animal'  is 
denied  of  something  man  may  exist  in  that  thing.  Suppose  it 
be  conceded  that  when  'animal'  is  denied,  man  may  persist; 
yet  it  was  formerly  conceded  that  'man'  necessarily  requires 
'animal',  viz.  in  the  consequence:  if  there  is  man  there  is 
animal.  And  so  it  is  contingent  that  what  is  not  animal,  be 
animal;  for  what  the  antecedent  admits,  the  consequent 
admits.  .  .  .  But  this  is  impossible.  .  .  . 

Quite  definitely  propositional  are  the  rules: 

1.08  Whatever  follows  from  the  consequent  (follows)  also 
from  the  antecedent; 

Whatever  implies  the  antecedent  (implies)  also  the  conse- 

used  in  the  derivation  of  categorical  syllogisms. 

While  it  is  clear  that  the  primary  source  of  all  this  doctrine  of 
consequence  is  the  De  Differentiis  Topicis  of  Boethius,  we  can  also 
see  the  germ  of  later  developments  in  Abelard's  realization  that  some 
are  deducible  from  others  (1.07,  1.09),  and  his  examinations  of  some 
that  he  finds  doubtful  (1.10). 

It  is  noteworthy  that  categorical  syllogisms  are  presented  entirely 
by  means  of  concrete  instances  and  metalogical  rules  (regulae)  — 
which  are  not  reckoned  as  maxims  since  the  inferential  (or  impli- 
cative) 'force'  of  the  premisses  is  derived  entirely  from  the  disposi- 
tion of  the  terms,  is,  in  Abelard's  terminology,  'complexional',  a 
term  preserved  in  Kilwardby.  Variables  of  the  object-language  are 
nowhere  used.  Indeed,  except  in  expositions  of  the  Boethian 
hypothetical  syllogisms,  the  only  place  we  find  variables  in  Abelard 
is  a  passage  where  he  introduces  a  simple  lettered  diagram  to  help 
the  intuition  in  an  original  argument: 

1.11  If  a  genus  was  always  to  be  divided  into  proximate 
species  or  proximate  differences,  every  division  of  a  genus 

*  Emending  De  Rijk's  possint  to  possit 



would  be  dichotomous  -  which  was  Boethius's  view.  .  .  .  But 
I  remember  having  an  objection  to  this  on  the  score  of  (the 
predicament  of)  relation.  .  .  .  This  will  be  more  easily  seen  if 
we  designate  the  members  of  the  predicament  by  letters  and 
distinguish  its  arrangement  by  a  figure  like  this. 


D  F    G  L 

If  now  C  and  D  were  mutually  related  on  the  one  hand, 
B  and  C  on  the  other,  since  B  is  prior  to  its  species  Z),  while 
D  is  together  with  (simul)  its  relative  C,  B  would  precede  C; 
so  that  B  would  precede  both  its  species  and  its  relative; 
hence  also  itself. 

There  follow  two  more  arguments  to  show  that  the  system  suppos- 
ed figured  stands  or  falls  entire  with  any  one  of  its  parts. 

There  is  no  suggestion  in  Abelard  that  syncategorematica, 
important  for  later  theory  of  consequences,  are  a  primary  concern 
of  logic,  the  purpose  of  which  he  states  as  follows: 

1.12  Logic  is  not  a  science  of  using  or  composing  arguments, 
but  of  discerning  and  estimating  them  rightly,  why  some  are 
valid,  others  invalid. 

But  he  is  puzzled  about  the  signification,  if  any,  of  syncategore- 
matica, and  refers  to  various  contemporary  views: 

1.13  Conjunctions  and  prepositions  ought  to  have  some 
signification  of  their  own.  .  .  .  What  concepts  are  designated  by 
expressions  of  this  kind,  it  is  not  easy  to  say.  .  .  .  Some  think 
that  such  expressions  have  sense  but  no  reference  (solos 
intelledus  generare,  nullamque  rem  subiedam  habere)  as  they 
grant  also  to  be  the  case  with  propositions.  .  .  .  There  are  also 
some  who  make  out  that  logicians  have  quite  removed  such 
expressions  from  the  class  of  significant  ones.  .  .  .  The  opinion 
I  favour  is  that  of  the  grammarians  who  make  contributions 
to  logic,  that  we  should  admit  them  as  significant,  but  should 
say  that  their  significance  lies  in  their  determining  certain 
properties  of  the  references  (res)  of  the  words  governed  by  the 
prepositions.  .  .  .  Conjunctions  too,  as  indicating  conjunction 
of  things,  determine  a  property  in  their  regard,  e.g.  when  I 
say:  'a  man  and  a  horse  runs',  by  the  conjunction  'and'  I 
unite  them  in  runing,  and  at  the  same  time  indicate  that  by 
the  'and'. 



The  emergence  of  a  logic  of  propositions  from  one  of  terms  is 
exemplified  in  the  rather  sophisticated,  and  disputed,  distinction 
between  propositional  and  term  connectives : 

1.14  It  is  to  be  remarked  that  while  disjunctive  connectives 
are  applied  to  the  terms  both  of  categorical  and  hypothetical 
propositions,  they  seem  to  have  a  different  sense  in  each.  .  .  . 
But  some  allow  no  difference  .  .  .  saying  that  there  is  the 
same  proposition  when  it  is  said:  'Socrates  is  healthy  or  (we/) 
sick'  and  when  it  is  said:  'Either  Socrates  is  healthy  or  (aut) 
he    is    sick',    reckoning    every    disjunctive    as    hypothetical. 

Again  of  temporal  propositions,  compounded  by  means  of  'when', 
which  Abelard  treats  as  conjunctives,  he  says: 

1.15  It  is  evident  in  temporals  that  we  should  not  estimate 
the  relation  of  consequence  according  to  any  force  in  the 
relationship  of  terms,  .  .  .  but  only  in  the  mutual  accompani- 
ment (of  the  components). 

And  again,  with  the  addition  of  truth-conditions: 

1.16  In  these  (temporals)  in  which  the  relation  of  conse- 
quence is  to  be  reckoned  nothing  else  than  coincidence  in 
time  .  .  .  provided  the  members  are  true,  people  concede  that 
the  consequence  is  true,  and  otherwise  false. 

Some  'rules'  follow  for  the  construction  of  consequences  on  this 
basis.  Among  them  the  following  deserves  special  notice : 

1.17  Of  whatever  (hypotheticals)  the  antecedents  are 
concomitant,  the  consequents  too  (are  concomitant),  thus: 
if  when  he  is  a  man  he  is  a  doctor,  when  he  is  an  animal  he  is 
an  artificer. 

This  is  Leibniz's  praeclarum  Iheorema  (cf.  43.37)  in  essentials, 
though  it  seems  impossible  to  say  whether  Abelard  envisaged  it  in 
its  Leibnizian  classical  or  its  Russellian  propositional  form.  He 
explains  it  indeed  by  saying  that  'as  "man"  is  necessarily  antecedent 
to  "animal",  so  "doctor"  is  to  "artificer"',  yet  he  clearly  thinks  of 
it  as  compounded  of  propositions.  The  fact  is  that  the  two  kinds  of 
logic  were  not  yet  perfectly  clearly  distinguished.  A  further  indication 
that  the  full  generality  of  propositional  logic  had  not  yet  been  achie- 
ved, though,  as  we  have  seen,  it  was  already  in  the  making,  is  that 
while  Abelard  gives  us  as  a  consequence:  if  he  is  man  and  stone,  he 
is  animal,  he  does  not  rise  to:  if  he  is  man  and  stone,  he  is  man. 

There  is  evidently  a  vast  deal  more  to  be  said  both  about  the 
prowess  and  the  limitations  of  this  logician,  both  on  this  and  other 
subjects,  but  we  already  exceed  the  limits  of  discussion  proper  to 
this  history.  It  is,  however,  certain  that  the  serious  beginnings  of 
Scholastic  logic  must  be  looked  for  in  the  12th  century. 




Grateful  acknowledgements  are  due  to  Bertrand  Russell  for 
permission  to  quote  from  his  The  Principles  of  Mathematics 
(London,  1903),  and  to  the  respective  publishers  for  permission 
to  use  passages  from:  The  Dialogues  of  Plato,  translated  in 
English  by  B.  Jowett  (Oxford  University  Press);  The  Works 
of  Aristotle,  translated  into  English  under  the  editorship  of 
J.  A.  Smith  andW.  D.Ross  (Oxford  University  Press) ;  Principia 
Mathematica,  by  A.  N.  Whitehead  and  Bertrand  Russell  (Cam- 
bridge University  Press,  1925-27) ;  Traciatus  Logico-Philo- 
sophicus,  by  L.Wittgenstein  (Routledge  and  Kegan  Paul,  1922) ; 
Translations  from  the  Philosophical  Writings  of  Gottlob  Frege, 
by  Peter  Geach  and  Max  Black  (Basil  Blackwell,  Oxford,  1952) ; 
to  The  Belknap  Press  of  Harvard  University  Press,  for  quota- 
tions from  The  Collected  Papers  of  Charles  Sanders  Peirce,  edited 
by  Charles  Hartshorne  and  Paul  Weiss  (copyright  1933  by  the 
President  and  Fellows  of  Harvard  University). 


Preface  to  the  German  edition v 

Translator's  preface  to  the  English  edition 

A.  General vii 

B.  Abelard viii 


§  1.    The  concept  of  formal  logic 2 

§  2.   On  the  history  of  the  history  of  logic 4 

A.  The  beginnings 4 

B.  Prejudices 4 

1.  Thomas  Reid 5 

2.  Kant 6 

3.  Prantl 6 

4.  After  Prantl 8 

C.  Research  in  the  20th  century 9 

§   3.    The  evolution  of  formal  logic 10 

A.  Concerning  the  geography  and  chrorretogy  of  logic    ...  10 

B.  How  logic  evolved 12 

C.  The  varieties  of  logic 12 

D.  The  unity  of  logic 14 

E.  The  problem  of  progress 15 

§  4,  Method  and  plan 18 

A.  History  of  problems,  and  documentation 18 

B.  Plan  of  the  work 18 

C.  Character  of  the  contents 19 

§   5.    Terminology 20 

A.  Technical  expressions 20 

B.  Concerning  mathematico-logical  symbolism         ....  22 

C.  Typographical  conventions 23 

The  Greek  Variety  of  Logic 

§   6.    Introduction  to  Greek  logic 26 

A.  Logicians  in  chronological  order 26 

B.  Periods '27 

C.  State  of  research 27 



I.  The  precursors 

§   7.    The  beginnings 29 

A.  Texts 29 

B.  Significance 31 

§   8.   Plato 33 

A.  Concept  of  logic 33 

B.  Approaches  to  logical  formulae 34 

C.  Diaeresis 35 

II.  Aristotle 

§   9.    The  work  of  Aristotle  and  the  problems  of  its  literary  history     .        .  40 

A.  Works 40 

B.  Problems 40 

1.  Authenticity 40 

2.  Character 41 

3.  Chronology 41 

§   10.   Concept  of  logic.  Semiotic 44 

A.  Name  and  place  of  logic 44 

B.  The  subject-matter  of  logic 44 

C.  Syntax 46 

D.  Semantics 47 

§   11.    The  topics 49 

A.  Subject  and  purpose 49 

B.  Predicables 51 

C.  Categories 53 

D.  Sophistic 54 

§   12.    Theory  of  opposition;  principle  of  contradiction;  principle  of  tertium 

exclusum 57 

A.  Theory  of  opposition 57 

B.  Obversion 59 

C.  The  principle  of  contradiction 60 

D.  The  principle  of  tertium  exclusum 62 

§   13.  Assertoric  syllogistic 63 

A.  Text 64 

B.  Interpretation 66 

C.  Structure  of  the  syllogism 69 

D.  The  figures  and  further  syllogisms 70 

§   14.  Axiomatization  of  the  syllogistic.  Further  laws 72 

A.  Axiomatic  theory  of  the  system 72 

B.  Systems  of  syllogistic 75 

C.  Direct  proof 76 

D.  Indirect  proof 77 

E.  Dictum  de  omni  et  nullo 79 

F.  Beginnings  of  a  metalogical  system 80 

G.  The  inventio  medii 80 

§  15.  Modal  logic 81 

A.  Modalities 81 

B.  Structure  of  modal  sentences 83 

C.  Negation  and  conversion 84 

D.  Syllogisms 85 

§  16.    Non-analytic  laws  and  rules 88 

A.  Two  kinds  of  inference 89 

B.  Laws  of  class-  and  predicate-logic 91 

C.  Theory  of  identity 92 

D.  Syllogisms  from  hypotheses 93 

E.  Laws  of  the  logic  of  relations 95 

F.  Propositional  rules  and  laws 97 

Summary 98 



§  17.    Theophrastus 99 

A.  Development  and  alteration  of  various  doctrines       .        .        .  99 

B.  Modal  logic 101 

C.  Hypothetical  syllogisms 103 

III.  The  Megarian-Stoic  School 

§   18.   Historical  survey 105 

A.  Thinkers  and  schools 105 

B.  Problems  of  literary  history 107 

C.  Origin  and  nature     .                         108 

§   19.   Concept  of  logic.  Semiotics.  Modalities 109 

A.  Logic 109 

B.  Lecta 110 

C.  Syntax Ill 

D.  Doctrine  of  categories 113 

E.  Truth 114 

F.  Modalities 114 

§   20.   Proposilional  functors 115 

A.  Negation 116 

B.  Implication 116 

1.  Philonian  implication 117 

2.  Diodorean  implication 117 

3.  'Connexive'  implication 118 

4.  'Inclusive' implication 119 

C.  Disjunction 119 

1.  Complete  disjunction 119 

2.  Incomplete  disjunction 120 

D.  Conjunction 121 

E.  Equivalence 121 

F.  Other  functors 121 

§   21.  Arguments  and  schemes  of  inference 122 

A.  Conclusive,  true  and  demonstrative  arguments  .        .        .        .  122 

B.  Non-syllogistic  arguments 124 

C.  Further  kinds  of  argument 124 

D.  Schemes  of  inference 125 

§  22.  Axiomatization.  Compound  arguments 126 

A.  The  indemonstrables 126 

B.  Metatheorems 127 

C.  Derivation  of  compound  arguments 128 

D.  Further  derived  arguments 130 

§   23.    The  liar 130 

A.  History 131 

B.  Formulation 131 

C.  Efforts  at  solution 132 

IV.  The   Close  of  Antiquity 

§   24.   Period  of  commentaries  and  handbooks 134 

A.  Characteristics  and  historical  survey 134 

B.  The  tree  of  Porphyry 135 

C.  Extension  of  logical  technique 136 

D.  Fresh  division  of  implication 137 

E.  Boethius's  hypothetical  syllogisms 139 

F.  Alterations  and  development  of  the  categorical  syllogistic       .  140 

G.  The  supposedly  fourth  figure 141 

H.   Pons  asinorum 143 

I.  Anticipation  of  the  logic  of  relations 144 

Summary 144 




The  Scholastic  Variety  of  Logic 

§  25.    Introduction  to  scholastic  logic 148 

A.  State  of  research 148 

B.  Provisional  periods 148 

C.  The  problem  of  sources 150 

D.  Logic  and  the  schools 150 

E.  Method 151 

F.  Characteristics 152 

I.  Semiotic  foundations 

§  26.  Subject-matter  of  logic 153 

A.  Basic  notions  of  semiotics 153 

B.  Logic  as  a  theory  of  second  intentions 154 

C.  Formal  logic  as  a  theory  of  syncategorematic  expressions        .  156 

D.  Content  of  the  works 159 

§   27.  Supposition 162 

A.  Concept  of  supposition 163 

B.  Material  and  formal  supposition 164 

C.  Simple  supposition 168 

D.  Personal  supposition 171 

E.  Interpretation  in  modern  terms 173 

§  28.  Ampliation,  appellation,  analogy 173 

A.  Ampliation 173 

B.  Appellation 175 

C.  Analogy 177 

§  29.  Structure  and  sense  of  propositions 180 

A.  Division  of  propositions 180 

B.  Analysis  of  propositions 180 

C.  Analysis  of  modal  propositions:  dictum  and  modus  .        .        .  182 

D.  Composite  and  divided  senses 184 

E.  Reference  of  propositions 187 

II.  Propositional   Logic 

§  30.   Notion  and  division  of  consequences 189 

A.  Historical  survey 189 

B.  Definition  of  consequence 190 

C.  Division  of  consequences 191 

D.  Meaning  of  implication 195 

E.  Disjunction 197 

§   31.   Propositional  consequences 198 

A.  Hypothetical  propositions 198 

B.  Kilwardby 198 

C.  Albert  of  Saxony 199 

D.  Paul  of  Venice 205 

E.  Rules  of  consequences  ut  nunc 208 

III.  Logic   of  Terms 

§   32.  Assertoric  syllogistic 210 

A.  Early  mnemonics 210 

B.  Barbara-Celarent 211 

C.  Barbara-Celaront 215 

D.  The  fourth  figure 216 

1.  Among  the  Latins 216 

2.  In  Albalag 217 

E.  Combinatorial  method 219 

F.  Inventio  medii,  pons  asinorum 219 



G.  The  problem  of  the  null  class 221 

1.  St.  Vincent  Ferrer 221 

2.  Paul  of  Venice 223 

3.  John  of  St.  Thomas 223 

§   33.   Modal  syllogistic 224 

A.  Albert  the  Great 224 

B.  Pseudo-Scotus 225 

C.  Ockham 227 

D.  Logic  of  propositions  in  future  and  past  tenses  ....  230 
§  34.  Other  formulae 231 

A.  Syllogisms  with  singular  terms 232 

B.  Analysis  of  'every'  and  'some' 234 

C.  Exponible  propositions 234 

D.  Oblique  syllogisms 236 

§   35.  Antinomies 237 

A.  Development 237 

B.  Formulation 239 

1.  The  liar 239 

2.  Other  antinomies 240 

C.  Solutions 241 

1.  The  first  twelve  solutions 241 

2.  The  thirteenth  solution 244 

3.  The  fourteenth  solution 246 

4.  Preliminaries  to  the  solution  of  Paul  of  Venice      .        .        .  247 

5.  The  solution  of  Paul  of  Venice 249 

Summary 251 

Transitional  Period 

§   36.    The  'classical'  logic 254 

A.  Humanism 254 

B.  Content 256 

C.  Psychologism 257 

D.  Leibniz 258 

E.  Comprehension  and  extension 258 

F.  The  fourth  figure  and  subaltern  moods 259 

G.  Syllogistic  diagrams 260 

H.   Quantification  of  the  predicate 262 

The  Mathematical  Variety  of  Logic 

I.  General   Foundations 

§37.    Introduction  to  mathematical  logic 266 

A.  Characteristics 266 

B.  Chronological  sequence 267 

C.  Frege 268 

D.  Periods 269 

E.  State  of  research 270 

F.  Method 271 

§   38.   Methods  of  mathematical  logic 272 

A.  Logical  calculus 272 

1.  Lull 272 

2.  Hobbes 273 

3.  Leibniz 274 



4.  Lambert 276 

5.  Gergonne 277 

6.  Boole 278 

7.  Peirce 279 

B.  Theory  of  proof 280 

1.  Bolzano 280 

2.  Frege 282 

C.  Metalogic 284 

§39.    The  concept  of  logic 286 

A.  The  logistic  position 287 

1.  Frege:  semantics 287 

2.  Frege:  logic  and  mathematics 289 

3.  Russell 290 

4.  Frege:  number 291 

B.  Formalism 292 

C.  Intuitionism 293 

II.  The   First   Period 

§   40.    The  Boolean  calculus 296 

A.  De  Morgan 296 

B.  Boole 298 

1.  Symbolism  and  basic  concepts 298 

2.  Applications 301 

C.  The  logical  sum 302 

D.  Inclusion 303 

E.  Peano 306 

III.  Propositional  Logic 

§   41.   Propositional  logic:  basic  concepts  and  symbolism       ....  307 

A.  Boole 307 

B.  McColl 309 

C.  Frege 310 

1.  Content  and  judgment 310 

2.  Implication 311 

D.  Peirce 313 

E.  Applications  of  his  symbolism  by  Frege 314 

F.  Negation  and  sum  in  Frege             316 

G.  Peano's  symbolism  for  propositional  logic 317 

H.  Later  development  of  symbolism  for  propositional  logic  .        .  318 

§  42.  Function,  variable,  truth-value 319 

A.  Logical  form 320 

B.  Concept  of  function:  Frege 320 

C.  Propositional  function:  Russell 322 

D.  Many-place  functions 323 

E.  The  variable 325 

1.  Frege 325 

2.  Russell 326 

F.  Truth-values 327 

G.  Truth-matrices 330 

1.  Peirce            330 

2.  Wittgenstein 331 

H.   Decision  procedure  of  Lukasiewicz 333 

§  43.   Propositional  logic  as  a  system 335 

A.  McColl 335 

B.  Frege's  rules  of  inference 337 

C.  Propositional  laws  from  the  Begriffsschrift 338 

D.  Whitehead  and  Russell 340 

1.   Primitive  symbols  and  definition 340 


2.  Axioms  (primitive  propositions) 340 

3.  Statement  of  proofs 341 

4.  Laws 342 

E.  Sheffer's  functor 344 

F.  Lukasiewicz's  statement  of  proofs 345 

IV.  Logic  of  Terms 

§   44.   Predicate  logic 347 

A.  Quantifiers 348 

1.  Mitchell 348 

2.  Peirce    .  348 

3.  Peano 349 

4.  Frege 350 

B.  Apparent  variables 353 

1.  Peano 353 

2.  Whitehead  and  Russell 353 

C.  Formal  implication 354 

D.  Laws  of  one-place  predicates 355 

E.  Laws  of  many-place  predicates 356 

F.  Identity 357 

§   45.    The  logic  of  classes 359 

A.  Individual  and  class.  Concept  of  element 360 

B.  Meaning  and  extension 360 

C.  The  plural  article 362 

D.  Definition  of  classes  by  functions 363 

E.  Product  and  inclusion  of  classes 364 

§   46.   Existence 365 

A.  The  null  class 365 

B.  Null  class  and  assertoric  syllogistic 366 

C.  Description 367 

1.  The  definite  article:  Frege 367 

2.  Logical  existence 369 

3.  Description  in  Russell 371 

4.  Symbolism 373 

(a)  Peano 373 

(b)  Principia 374 

V.  Other   Doctrines 

§   47.  Logic  of  relations 375 

A.  Laying  the  foundations 375 

1.  De  Morgan 375 

2.  Peirce 377 

3.  Russell 379 

4.  Principia 380 

B.  Series 3S4 

1.  Frege 385 

2.  Principia 385 

C.  Isomorphy 386 

§   48.  Antinomies  and  theories  of  types 387 

A.  Historical  survey 3S7 

B.  The  antinomies 388 

C.  Anticipations  of  the  theory  of  types 391 

D.  The  ramified  theory  of  types 393 

E.  Systematic  ambiguity 396 

F.  The  axiom  of  reducibility 398 

G.  Simple  theory  of  types 399 

1.  Chwistek 399 

2.  Ramsey 400 



§  49.  Some  recent  doctrines 402 

A.  Strict  implication:  Lewis 403 

B.  Many-valued  logics:  Lukasiewicz 405 

C.  Godel's  theorem 407 

D.  Combinatory  logic 411 

Summary 412 

The  Indian  Variety  of  Logic 

§  50.    Introduction  to  Indian  logic 416 

A.  Historical  survey 416 

B.  Evolution  of  formal  logic 417 

C.  State  of  research 419 

D.  Method 420 

§   51.    The  precursors 421 

A.  Milinda-Panha 421 

B.  Kathavatthu 421 

C.  The  ten-membered  formula 423 

§  52.    Vaisesika-  and  Nyaya-sutra 425 

A.  Vaisesika-sutra 425 

1.  Doctrine  of  categories 425 

2.  Inference 426 

B.  Nyaya-sutra 426 

1.  Text 426 

2.  Vatsyayana's  commentary 428 

3.  Interpretation 429 

§  53.    The  rise  of  formal  logic .  431 

A.  Main  stages  of  development 431 

B.  Terminology 432 

C.  The  three-membered  syllogism 433 

D.  The  three-membered  rule:  trairupya 435 

E.  Wheel  of  reasons:  hetu-cakra 435 

F.  'Eva' 437 

G.  Universal  connection 437 

H.   Final  form  of  the  doctrine 439 

§   54.  Some  other  logical  doctrines 440 

A.  Apoha 441 

B.  Definitions  of  vyapti 441 

C.  Some  basic  concepts 442 

D.  The  law  of  double  negation 444 

E.  Relation  logic,  definition  of  number 445 

Summary 446 


I.  References 451 

II.  Bibliography 460 

III.  Indices 535 

1.  Index  of  names 535 

2.  Index  of  logical  symbols 541 

3.  Index  of  mnemonics 544 

4.  Subject  index 545 

Plate  I opposite  p.  220 

Plate  II opposite  p.  260 

Plate  III  opposite  p. 274 

Plate  IV opposite  p.  316 




1    Bochenski,  Formal  Logic 


Preliminary  definition  of  the  subject  matter  of  the  history  of 
logic  is  hard  to  come  by.  For  apart  from  'philosophy'  there  is  perhaps 
no  name  of  a  branch  of  knowledge  that  has  been  given  so  many 
meanings  as  'logic'.  Sometimes  the  whole  of  philosophy,  and  even 
knowledge  in  general,  has  been  thus  named,  from  metaphysics  on 
the  one  hand,  cf.  Hegel,  to  aesthetics  ('logic  of  beauty')  on  the  other, 
with  psychology,  epistemology,  mathematics  etc.  in  between.  With 
such  a  wide  choice  it  is  quite  impossible  to  include  in  a  history  of 
logical  problems  all  that  has  been  termed  'logic'  in  the  course  of 
western  thought.  To  do  so  would  practically  involve  writing  a 
general  history  of  philosophy. 

But  it  does  not  follow  that  the  use  of  the  name  'logic'  must  be 
quite  arbitrary,  for  history  provides  several  clues  to  guide  a  choice 
between  its  many  meanings.  This  choice  can  be  arrived  at  by  the 
following  stages. 

1.  First  let  us  discard  whatever  most  authors  either  expressly 
ascribe  to  some  other  discipline,  or  call  'logic'  with  the  addition  of 
an  adjective,  as  for  example  epistemology,  transcendental  logic, 
ontology  etc. 

2.  When  we  examine  what  remains,  we  find  that  there  is  one 
thinker  who  so  distinctly  marked  out  the  basic  problems  of  this 
residual  domain  that  all  later  western  inquirers  trace  their  descent 
from  him:  Aristotle.  Admittedly,  in  the  course  of  centuries  very 
many  of  these  inquirers  -  among  them  even  his  principal  pupil  and 
successor  Theophrastus  -  have  altered  Aristotelian  positions  and 
replaced  them  with  others.  But  the  essential  problematic  of  their  work 
was,  so  far  as  we  know,  in  constant  dependence  in  one  way  or  another 
on  that  of  Aristotle's  Organon.  Consequently  we  shall  denote  as 
'logic'  primarily  those  problems  which  have  developed  from  that 

3.  When  we  come  to  the  post-Aristotelian  history  of  logic,  we  can 
easily  see  that  one  part  of  the  Organon  has  exercised  the  most 
decisive  influence,  namely  the  Prior  Analytics.  At  some  periods 
other  parts  too,  such  as  the  Topics  or  the  Posterior  Analytics,  have 
indeed  been  keenly  investigated  and  developed.  But  it  is  generally 
true  of  all  periods  marked  by  an  active  interest  in  the  Organon  that 
the  problems  mainly  discussed  are  of  the  kind  already  to  hand  in 
the  Prior  Analytics.  So  the  third  step  brings  us  to  the  point  of 
describing  as  'logic'  in  the  stricter  sense  that  kind  of  problematic 
presented  in  the  Prior  Analytics. 

4.  The  Prior  Analytics  treats  of  the  so-called  syllogism,  this  being 
defined  as  aXoyo<;in  which  if  something  is  posited,  something  else 
necessarily  follows.  Moreover  such  Xoyot,  are  there  treated  as 
formulas  which  exhibit  variables  in  place  of  words  with  constant 


meaning;  an  example  is  'B  belongs  to  all  A'.  The  problem  evidently, 
though  not  explicitly,  presented  by  Aristotle  in  this  epoch-making 
work,  could  be  formulated  as  follows.  What  formulas  of  the  prescri- 
bed type,  when  their  variables  are  replaced   by  constants,   yield 

conditional  statements  such  that  when  the  antecedent  is  accepted, 
the  consequent  must  be  admitted?  Such  formulas  are  called  'logical 
sentences'.  We  shall  accordingly  treat  sentences  of  this  kind  as  a 
principal  subject  of  logic. 

5.  Some  logicians  have  limited  themselves  to  the  discovery, 
examination  and  systematic  ordering  of  logical  theorems,  e.g.  many 
scholastic  and  mathematical  logicians,  as  also  Aristotle  himself  in 
the  Prior  Analytics.  But  logic  so  understood  seems  too  narrowly 
conceived.  For  two  kinds  of  problem  naturally  arise  out  of  the 
theorems.  First  those  about  their  nature  -  are  they  linguistic 
expressions,  word-structures,  psychical  forms  or  functions,  objective 
complexes?  What  does  a  logical  law  mean,  what  does  a  statement 
mean?  These  are  problems  which  nowadays  are  dealt  with  in  semio- 
tics. Second,  problems  relevant  to  the  question  how  logical  laws 
can  be  correctly  applied  to  practical  scientific  thought.  These  were 
dealt  with  by  Aristotle  himself,  principally  in  the  Posterior  Analytics, 
and  nowadays  are  the  concern  of  general  methodology.  So  semiotic 
and  methodological  problems  are  closely  connected  with  logic; 
in  practice  they  are  always  based  on  semiotics  and  completed  in 
methodology.  What  remains  over  and  above  these  two  disciplines 
we  shall  call  formal  logic. 

6.  A  complete  history  of  the  problems  of  logic  must  then  have 
formal  logic  at  its  centre,  but  treat  also  of  the  development  of 
problems  of  semiotics  and  methodology.  Before  all  else  it  must  put 
the  question :  what  problems  were  in  the  past  posited  with  reference 
to  the  formulation,  assessment  and  systematization  of  the  laws  of 
formal  logic?  Beyond  that  it  must  look  for  the  sense  in  which  these 
problems  were  understood  by  the  various  logicians  of  the  past,  and 
also  attempt  to  answer  the  question  of  the  application  of  these 
laws  in  scientific  practice.  We  have  now  delimited  our  subject,  and 
done  so,  as  we  think,  in  accordance  with  historical  evidence. 

But  such  a  program  has  proved  to  be  beyond  accomplishment. 
Not  only  is  our  present  knowledge  of  seim'otic  and  methodological 
questions  in  the  most  important  periods  too  fragmentary,  but  even 
where  the  material  is  sufficiently  available,  a  thorough  treatment 
would  lead  too  far  afield.  Accordingly  we  have  resolved  to  limit 
ourselves  in  the  main  to  matters  of  purely  formal  logic,  giving  only 
incidental  consideration  to  points"  from  the  other  domains. 

Thus  the  subject  of  this  work  is  constituted  by  those  problems 
which  are  relevant  to  the  structure,  interconnection  and  truth  of 
sentences  of  formal  logic  (similar  to  the  Aristotelian  syllogism). 
Does  it  or  does  it  not  follow?  And,  why?  How  can  one  prove  the 


validity  of  this  or  that  sentence  of  formal  logic?  How  define  one 
or  another  logical  constant,  e.g.  'or',  'and',  'if--then',  'every'  etc. 
Those  are  the  questions  of  which  the  history  will  here  be  considered. 



The  first  efforts  to  write  a  history  of  logic  are  to  be  found  among 
the  humanists,  and  perhaps  Petrus  Ramus  may  here  be  counted  as 
the  first  historian.  In  his  Scholarum  dialeclicarum  libri  XX  we  find 
some  thirty  long  colums  allotted  to  this  history.  To  be  sure,  Ramus's 
imagination  far  outruns  his  logic:  he  speaks  of  a  logica  Patrum 
in  which  Noah  and  Prometheus  figure  as  the  first  logicians, 
then  of  a  logica  mathematicorum  which  alludes  to  the  Pythagoreans. 
There  follow  a  logica  physicorum  (Zeno  of  Elea,  Hippocrates,  Demo- 
critus  etc.),  the  logica  Socratis,  Pyrrhonis  el  Epicrelici  (sicl),  the 
logica  Anlislheniorum  el  Stoicorum  (here  the  Megarians  too  are 
named,  among  others  Diodorus  Cronus)  and  the  logica  Academiorum. 
Only  then  comes  the  logica  Peripaleticorum  where  Ramus  mentions 
what  he  calls  the  Arislotelis  bibliotheca  i.e.  the  Organon  (which 
according  to  him,  as  in  our  own  time  according  to  P.  Zurcher.  S.J.,  is 
not  by  Aristotle),  and  finally  the  logica  Aristoteleorum  inter prelum  el 
praecipue  Galeni  (2.01). 

This  book  was  written  in  the  middle  of  the  16th  century.  Some 
fifty  years  later  we  find  a  less  comprehensive  but  more  scientific 
attempt  by  B.  Keckermann.  His  work  (2.02)  is  still  valuable,  parti- 
cularly for  a  large  collection  of  accurately  dated  titles.  It  remains  an 
important  foundation  for  the  study  of  lGth  century  logic.  But  its 
judgments  are  not  much  more  reliable  than  those  of  Ramus.  Kecker- 
mann seems  to  have  given  only  a  cursory  reading  to  most  of  the 
logicians  he  cites,  Hospinianus  (2.03)  for  example.  The  book  is 
indeed  more  of  a  bibliography  than  a  history  of  logic. 


For  all  his  faults,  Ramus  was  a  logician;  Keckermann  too  had 
some  knowledge  of  the  subject.  The  same  can  seldom  be  said  of  their 
successors  until  Bolzano,  Peirce  and  Peano.  Most  historians  of  logic 
in  the  17th,  18th  and  19th  centuries  treat  of  ontological,  epistemolo- 
gical  and  psychological  problems  rather  than  of  logical  ones.  Further- 
more, everything  in  this  period,  with  few  exceptions,  is  so  condi- 
tioned by  the  then  prevailing  prejudices  that  we  may  count  the 
whole  period  as  part  of  the  pre-history  of  our  science. 


These  prejudices  are  essentially  three: 

1.  First,  everyone  was  convinced  that  formalism  has  very  little 
to  do  with  genuine  logic.  Hence  investigations  of  formal  logic  either 
passed  unnoticed  or  were  contemptuously  treated  as  quite  subsidiary. 

2.  Second,  and  in  part  because  of  the  prejudice  already  mentioned, 
the  scholastic  period  was  treated  as  a  media  tempestas,  a  'dark 
middle  age'  altogether  lacking  in  science.  But  as  the  Scholastics 
were  in  possession  of  a  highly  developed  formal  logic,  people  sought 
in  history  either  for  quite  different  'logics'  from  that  of  Aristotle 
(not  only  those  of  Noe  and  Epictetus,  as  Ramus  had  done,  but  even, 
after  his  time,  that  of  Ramus  himself),  or  at  least  for  a  supposedly 
better  interpretation  of  him,  which  put  the  whole  investigation  on  the 
wrong  track. 

3.  Finally,  of  equal  influence  was  a  strange  belief  in  the  linear 
development  of  every  science,  logic  included.  Hence  there  was  a 
permanent  inclination  to  rank  inferior  'modern'  books  higher  than 
works  of  genius  from  older  classical  writers. 

1.  Thomas  Reid 

As  an  example  of  how  history  was  written  then,  we  shall  cite  one 
man  who  had  the  good  will  at  least  to  read  Aristotle,  and  who 
succeeded  in  doing  so  for  most  parts  of  the  Organon,  though  he 
failed  for  just  the  most  important  treatises.  Here  are  his  own  words 
on  the  subject: 

2.04  In  attempting  to  give  some  account  of  the  Analytics 
and  of  the  Topics  of  Aristotle,  ingenuity  requires  me  to  con- 
fess, that  though  I  have  often  purposed  to  read  the  whole 
with  care,  and  to  understand  what  is  intelligible,  yet  my 
courage  and  patience  always  failed  before  I  had  done.  Why 
should  I  throw  away  so  much  time  and  painful  attention 
upon  a  thing  of  so  little  real  use  ?  If  I  had  lived  in  those  ages 
when  the  knowledge  of  Aristotle's  Organon  entitled  a  man  to 
the  highest  rank  in  philosophy,  ambition  might  have  induced 
me  to  employ  upon  it  some  years'  painful  study;  and  less,  I 
conceive,  would  not  be  sufficient.  Such  reflections  as  these, 
always  got  the  better  of  my  resolution,  when  the  first  ardour 
began  to  cool.  All  I  can  say  is,  that  I  have  read  some  parts  of 
the  different  books  with  care,  some  slightly,  and  some  perhaps 
not  at  all.  ...  Of  all  reading  it  is  the  most  dry  and  the  most 
painful,  employing  an  infinite  labour  of  demonstration,  about 
things  of  the  most  abstract  nature,  delivered  in  a  laconic 
style,  and  often,  I  think,  with  affected  obscurity;  and  all  to 
prove  general  propositions,  which  when  applied  to  particular 
instances  appear  self-evident. 


In  the  first  place  this  is  a  really  touching  avowal  that  Reid 
lectured  on  the  teaching  of  a  logician  whom  he  had  not  once  read 
closely,  and,  what  is  much  more  important,  that  for  this  Scottish 
philosopher  formal  logic  was  useless,  incomprehensible  and  tedious. 
But  beyond  that,  the  texts  that  seemed  to  him  most  unintelligible 
and  useless  are  just  those  that  every  logician  counts  among  the  most 
exquisite  and  historically  fruitful. 

Nearly  all  philosophers  of  the  so-called  modern  period,  from  the 
humanists  to  the  rise  of  mathematical  logic,  held  similar  views.  In 
such  circumstances  there  could  be  no  scientific  history  of  logic,  for 
that  presupposes  some  understanding  of  the  science  of  logic. 

The  attitude  towards  formal  logic  just  described  will  be  further 
illustrated  in  the  chapter  on  'classical'  logic.  Here  we  shall  delay 
only  on  Kant,  who  expressed  opinions  directly  relevant  to  the 
history  of  logic. 

2.  Kanl 

Kant  did  not  fall  a  victim  to  the  first  and  third  of  the  prejudices 
just  mentioned.  He  had  the  insight  to  state  that  the  logic  of  his 
time  —  he  knew  no  other  —  was  no  better  than  that  of  Aristotle,  and 
went  on  to  draw  the  conclusion  that  logic  had  made  no  progress 
since  him. 

2.05  That  Logic  has  advanced  in  this  sure  course,  even  from 
the  earliest  times,  is  apparent  from  the  fact  that  since  Ari- 
stotle, it  has  been  unable  to  advance  a  step,  and  thus  to  all 
appearance  has  reached  its  completion.  For  if  some  of  the 
moderns  have  thought  to  enlarge  its  domain  by  introducing 
psychological  discussions  . . .  metaphysical  ...  or  anthropological 
discussions  . . .  this  attempt,  on  the  part  of  these  authors, 
only  shows  their  ignorance  of  the  peculiar  nature  of  logical 
science.  We  do  not  enlarge,  but  disfigure  the  sciences  when  we 
lose  sight  of  their  respective  limits,  and  allow  them  to  run  into 
one  another.  Now  logic  is  enclosed  within  limits  which  admit 
of  perfectly  clear  definition;  it  is  a  science  which  has  for  its 
object  nothing  but  the  exposition  and  proof  of  the  formal  laws 
of  all  thought,  whether  it  be  a  priori  or  empirical.  .  .  . 

3.  Prantl 

It  is  a  remarkable  fact,  unique  perhaps  in  the  writing  of  history, 
that  Carl  Prantl,  the  first  to  write  a  comprehensive  history  of 
western  logic  (2.06),  on  which  task  he  spent  a  lifetime,  did  it  pre- 
cisely to  prove  that  Kant  was  right,  i.e.  that  formal  logic  has  no 
history  at  all. 

His  great  work  contains  a  collection  of  texts,  often  arranged  from 



a  wrong  standpoint,  and  no  longer  sufficient  but  still  indispensable. 
He  is  the  first  to  take  and  discuss  seriously  all  the  ancient  and  scho- 
lastic logicians  to  whom  he  had  access,  though  mostly  in  a  polemi- 
cal and  mistaken  spirit.  Hence  one  can  say  that  he  founded  the 
history  of  logic  and  bequeathed  to  us  a  work  of  the  highest  utility. 

Yet  at  the  same  time  nearly  all  his  comments  on  these  logicians 
are  so  conditioned  by  the  prejudices  we  have  enumerated,  are  written 
too  with  such  ignorance  of  the  problems  of  logic,  that  he  cannot  be 
credited  with  any  scientific  value.  Prantl  starts  from  Kant's  asser- 
tion, believing  as  he  does  that  whatever  came  after  Aristotle  was 
only  a  corruption  of  Aristotle's  thought.  To  be  formal  in  logic,  is  in 
his  view  to  be  unscientific.  Further,  his  interpretations,  even  of 
Aristotle,  instead  of  being  based  on  the  texts,  rely  only  on  the 
standpoint  of  the  decadent  'modern'  logic.  Accordingly,  for  example, 
Aristotelian  syllogisms  are  misinterpreted  in  the  sense  of  Ockham, 
every  formula  of  propositional  logic  is  explained  in  the  logic  of 
terms,  investigation  of  objects  other  than  syllogistic  characterized  as 
'rank  luxuriance',  and  so  of  course  not  one  genuine  problem  of 
formal  logic  is  mentioned. 

While  this  attitude  by  itself  makes  the  work  wholly  unscientific 
and,  except  as  a  collection  of  texts,  worthless,  these  characteristics 
are  aggravated  by  a  real  hatred  of  all  that  Prantl,  owing  to  his 
logical  bias,  considers  incorrect.  And  this  hatred  is  extended  from 
the  teachings  to  the  teachers.  Conspicuous  among  its  victims  are  the 
thinkers  of  the  Megarian,  Stoic  and  Scholastic  traditions.  Ridicule, 
and  even  common  abuse,  is  heaped  on  them  by  reason  of  just  those 
passages  where  they  develop  manifestly  important  and  fruitful 
doctrines  of  formal  logic. 

We  shall  illustrate  this  with  some  passages  from  his  Geschichte  der 
Logik,  few  in  number  when  compared  with  the  many  available. 

Ghrysippus,  one  of  the  greatest  Stoic  logicians,  'really  accom- 
plished nothing  new  in  logic,  since  he  only  repeats  what  the  Peri- 
patetics had  already  made  available  and  the  peculiarities  introduced 
by  the  Megarians.  His  importance  consists  in  his  sinking  to  handle  the 
material  with  a  deplorable  degree  of  platitude,  triviality  and  scho- 
lastic niggling',  Chrysippus  'is  a  prototype  of  all  pedantic  narrow- 
mindedness'  (2.07).  Stoic  logic  is  in  general  a  'corruption'  of  that 
previously  attained  (2.08),  a  'boundless  stupidity',  since  'Even  he 
who  merely  copies  other  people's  work,  thereby  runs  the  risk  of 
bringing  to  view  only  his  own  blunders'  (2.09).  The  Stoic  laws  are 
'proofs  of  poverty  of  intellect'  (2.10).  And  the  Stoics  were  not  only 
stupid;  they  were  also  morally  bad  men,  because  they  were  subtle: 
their  attitude  has  'not  only  no  logical  worth,  but  in  the  realm  of 
ethics  manifests  a  moment  deficient  in  morality'  (2.11).  —  Of 
Scholasticism  Prantl  says :  'A  feeling  of  pity  steals  over  us  when  we 
see  how  even  such  partialities  as  are  possible  within  an  extremely 


limited  field  of  view  are  exploited  with  plodding  industry  even  to  the 
point  of  exhaustion,  or  when  centuries  are  wasted  in  their  fruitless 
efforts  to  systematize  nonsense'  (2.12).  Consequently  'so  far  as 
concerns  the  progress  of  every  science  that  can  properly  be  termed 
philosophy,  the  Middle  Ages  must  be  considered  as  a  lost  millennium' 

In  the  13th  century  and  later,  things  are  no  better.  'Between  the 
countless  authors  who  without  a  single  exception  subsist  only  on  the 
goods  of  others,  there  is  but  one  distinction  to  be  made.. There  are 
the  imbeciles  such  as  e.g.  Albertus  Magnus  and  Thomas  Aquinas, 
who  hastily  collect  ill-assorted  portions  of  other  people's  wealth  in  a 
thoughtless  passion  for  authority;  and  those  others,  such  as  e.g. 
Duns  Scotus,  Occam  and  Marsilius,  who  at  least  understand  with 
more  discernment  how  to  exploit  the  material  at  hand'  (2.14). 
'Albertus  Magnus  too  .  .  .  was  a  muddle-head'  (2.15).  To  take 
Thomas  Aquinas  'for  a  thinker  in  his  own  right'  would  be  'a  great 
mistake'  (2.16).  His  pretended  philosophy  is  only  'his  unintelligent 
confusion  of  two  essentially  different  standpoints;  since  only  a 
muddled  mind  can  .  .  .'  etc.  (2.17). 

Similar  judgment  is  passed  on  later  scholastic  logic;  a  chapter  on 
the  subject  is  headed  'Rankest  Luxuriance'  (2.18).  Prantl  regrets 
having  to  recount  the  views  of  these  logicians,  'since  the  only  alter- 
native interpretation  of  the  facts,  which  would  consist  simply  in 
saying  that  this  whole  logic  is  a  mindless  urge,  would  be  blameworthy 
in  a  historian,  and  without  sufficient  proof  would  not  gain  credence' 

To  refute  Prantl  in  detail  would  be  a  huge  and  hardly  profitable 
task.  It  is  better  to  disregard  him  entirely.  He  must,  unhappily,  be 
treated  as  non-existent  by  a  modern  historian  of  logic.  Refutation  is 
in  any  case  effected  by  the  total  results  of  subsequent  research  as 
recapitulated  in  this  book. 

4.  After  Prantl 

Prantl  exercised  a  decisive  influence  on  the  writing  of  history 
of  logic  in  the  19th  and  to  some  extent  also  in  the  20th  century.  Till 
the  rise  of  the  new  investigations  deriving  from  circles  acquainted 
with  mathematical  logic,  Prantl's  interpretations  and  evaluations 
were  uncritically  accepted  almost  entire.  For  the  most  part,  too,  the 
later  historians  of  logic  carried  still  further  than  Prantl  the  mingling 
of  non-logical  with  logical  questions.  This  can  be  seen  in  the  practice 
of  giving  a  great  deal  of  space  in  their  histories  to  thinkers  who  were 
not  logicians,  and  leaving  logicians  more  and  more  out  of  account. 

Some  examples  follow.  F.  Ueberweg,  himself  no  mean  logician, 
(he  could,  e.g.  distinguish  propositional  from  term-logic,  a  rare  gift 
in  the  19th  century),  devoted  four  pages  of  his  survey  of  the  history 
of  logic   (2.20)   to  Aristotle,   two  to   'the  Epicureans,   Stoics  and 



Sceptics',  two  to  the  whole  of  Scholasticism  -  but  fifty-five  to  the 
utterly  barren  period  stretching  from  Descartes  to  his  own  day. 
Therein  Schleiermacher,  for  instance,  gets  more  space  than  the 
Stoics,  and  Descartes  as  much  as  all  Scholasticism.  R.  Adamson 
(2.21)  allots  no  less  that  sixteen  pages  to  Kant,  but  only  five  to 
the  whole  period  between  the  death  of  Aristotle  and  Bacon,  com- 
prising the  Megarians,  Stoics,  Commentators  and  Scholastics.  A  few 
years  ago  Max  Polenz  gave  barely  a  dozen  pages  to  Stoic  logic  in  his 
big  book  on  this  school  (22.2). 

Along  with  this  basic  attitude  went  a  misunderstanding  of  ancient 
logical  teaching.  It  was  consistently  treated  as  though  exhibiting 
nothing  what  corresponded  with  the  content  of  'classical'  logic;  all 
else  either  went  quite  unnoticed  or  was  interpreted  in  the  sense  of 
the  'classical'  syllogistic,  or  again,  written  off  as  mere  subtlety.  It  is 
impossible  to  discuss  the  details  of  these  misinterpretations,  but  at 
least  they  should  be  illustrated  by  some  examples. 

The  Aristotelian  assertoric  syllogistic  is  distortedly  present  in 
'classical'  i.e.  Ockhamist  style  (34.01)  as  a  rule  of  inference  with  the 
immortal  'Socrates'  brought  into  the  minor  premiss,  whereas  for 
Aristotle  the  syllogism  is  a  conditional  propositional  form  (§13) 
without  any  singular  terms.  Stoic  logic  was  throughout  absurdly 
treated  as  a  term  logic  (2.23),  whereas  it  was  quite  plainly  a  pro- 
positional  logic  (§20).  Aristotelian  modal  logic  (§15)  was  so  little 
understood  that  when  A.  Becker  gave  the  correct  interpretation  of  its 
teaching  in  1934  (2.24),  his  view  was  generally  thought  to  be  revo- 
lutionary, though  in  essence  this  interpretation  is  quite  elementary 
and  was  known  to  Albert  the  Great  (33.03).  Aristotle  and  Thomas 
Aquinas  were  both  credited  with  the  Theophrastan  analysis  of 
modal  propositions  and  modal  syllogisms,  which  they  never  advo- 
cated (2.25). 

No  wonder  then  that  with  the  rise  of  mathematical  logic  theorems 
belonging  to  the  elementary  wealth  of  past  epochs  were  saddled  with 
the  names  of  De  Morgan,  Peirce  and  others;  there  was  as  yet  no 
scientific  history  of  formal  logic. 


Scientific  history  of  formal  logic,  free  from  the  prejudices  we  have 
mentioned  and  based  on  a  thorough  study  of  texts,  first  developed  in 
the  20th  century.  The  most  important  researches  in  the  various 
fields  are  referred  to  in  the  relevant  parts  of  our  survey.  Here  we 
shall  only  notice  the  following  points. 

The  rise  of  modern  history  of  logic  concerning  all  periods  save 
the  mathematical  was  made  possible  by  the  work  of  historians 
of  philosophy  and  philologists  in  the  19th  century.  These  published 
for  the  first  time  a  series  of  correct  texts  edited  with  reference  to 
their  context  in  the  history  of  literature.  But  the  majority  of  ancient 



philologists,  medievalists  and  Sanskrit  scholars  had  only  slight 
understanding  of  and  little  interest  in  formal  logic.  History  of 
logic  could  not  be  established  on  the  sole  basis  of  their  great  and 
laborious  work. 

For  its  appearance  we  have  to  thank  the  fact  that  formal  logic 
took  on  a  new  lease  of  life  and  was  re-born  as  mathematical.  Nearly 
all  the  more  recent  researches  in  this  history  were  carried  out  by 
mathematical  logicians  or  by  historians  trained  in  mathematical 
logic.  We  mention  only  three  here:  Charles  Sanders  Peirce,  the  fore- 
runner of  modern  research,  versed  in  ancient  and  scholastic  logic; 
Heinrich  Scholz  and  Jan  Lukasiewicz,  with  their  publications  of 
1931  and  1935  (2.26,  2.27),  both  exercising  a  decisive  influence 
on  many  parts  of  the  history  of  logic,  thanks  to  whom  there  have 
appeared  serious  studies  of  ancient,  medieval  and  Indian  logic. 

But  still  we  have  only  made  a  start.  Though  we  are  already  in 
possession  of  basic  insights  into  the  nature  of  the  different  historical 
varieties  of  formal  logic,  our  knowledge  is  still  mostly  fragmentary. 
This  is  markedly  the  case  for  Scholastic  and  Indian  logic.  But  as 
the  history  of  logic  is  now  being  systematically  attended  to  by  a 
small  group  of  researchers  it  can  be  foreseen  that  this  state  of 
affairs  will  be  improved  in  the  coming  decades. 


As  an  introduction  to  the  present  state  of  research  and  to  justify 
the  arrangement  of  this  book,  a  summary  presentation  of  results  is 
now  needed.  The  view  we  present  is  a  new  one  of  the  growth  of 
formal  logic,  stated  here  for  the  first  time.  It  is  a  view  which  markedly 
diverges  not  only  from  all  previous  conceptions  of  the  history  of 
logic,  but  also  from  opinions  that  are  still  widespread  about  the 
general  history  of  thought.  But  it  is  no  'synthetic  a  priori  judgment', 
rather  is  it  a  position  adopted  in  accordance  with  empirical  findings 
and  based  on  the  total  results  of  the  present  book.  Its  significance 
seems  not  to  be  confined  within  the  boundaries  of  the  history  of 
logic:  the  view  might  be  taken  as  a  contribution  to  the  general 
history  of  human  thought  and  hence  to  the  sociology  of  knowledge. 


Formal  logic,  so  far  as  we  know,  originated  in  two  and  only  two 
cultural  regions:  in  the  west  and  in  India.  Elsewhere,  e.g.  in  China, 
we  do  occasionally  find  a  method  of  discussion  and  a  sophistic 
(3.01),  but  no  formal  logic  in  the  sense  of  Aristotle  or  Dignaga  was 
developed  there. 

Both  these  logics  later  spread  far  beyond  the  frontiers  of  their 
native  region.  We  are  not  now  speaking  merely  of  the  extension  of 



European  logic  to  America,  Australia  and  other  countries  settled 
from  Europe;  North  America,  for  instance,  which  from  the  time 
of  Peirce  has  been  one  of  the  most  important  centres  of  logical 
research,  can  be  treated  as  belonging  to  the  western  cultural  region. 
Rather  it  is  a  matter  of  western  logic  having  conquered  the  Arabian 
world  in  the  high  Middle  Age,  and  penetrated  Armenian  culture 
through  missionaries.*  Other  examples  could  be  adduced.  The 
same  holds  for  Indian  logic,  which  penetrated  to  Tibet,  China, 
Japan  and  elsewhere.  Geographically,  then,  we  arc  concerned  with 
two  vital  centres  of  evolution  for  logic,  whose  influence  eventually 
spread  far  abroad. 

On  the  subject  of  the  chronology  of  logic  and  its  division  into 
periods  there  is  this  to  be  said :  this  history  begins  in  Europe  in  the 
4th  century  B.C.,  in  India  about  the  1st  century  a.d.  Previously 
there  is  in  Greece,  India  and  China,  perhaps  also  in  other  places, 
something  like  a  pre-history  of  logic;  but  it  is  a  complete  mistake 
to  speak  of  a  'logic  of  the  Upanishads'  or  a  'logic  of  the  Pythagoreans'. 
Thinkers  of  these  schools  did  indeed  establish  chains  of  inference, 
but  logic  consists  in  studying  inference,  not  in  inferring.  No  such 
study  can  be  detected  with  certainty  before  Plato  and  the  Nyaga; 
at  best  we  find  some  customary,  fixed  and  canonical  rules  of  dis- 
cussion, but  any  complete  critical  appreciation  and  analysis  of 
these  rules  are  missing. 

The  history  of  western  logic  can  be  divided  into  five  periods: 
1.  the  ancient  period  (to  the  6th  century  a.d.);  2.  the  high  Middle 
Age  (7th  to  11th  centuries);  3.  the  Scholastic  period  (11th  to  loth 
centuries);  4.  the  older  period  of  modern  'classical'  logic  (16th  to 
19th  centuries);  5.  mathematical  logic  (from  the  middle  of  the  19th 
century).  Two  of  those  are  not  creative  periods  —  the  high  Middle 
Age  and  the  time  of  'classical'  logic,  so  that  they  can  be  left  almost 
unnoticed  in  a  history  of  problems.  The  hypothesis  that  there  was 
no  creative  logical  investigation  between  the  ancient  and  Scholastic 
periods  might  very  probably  be  destroyed  by  a  knowledge  of  Arabian 
logic,  but  so  far  little  work  has  been  done  on  this,  and  as  the  results 
of  what  research  has  been  undertaken  are  only  to  be  found  in  Arabic, 
they  are  unfortunately  not  available  to  us. 

Indian  logic  cannot  so  far  be  divided  into  periods  with  comparable 
exactness.  It  only  seems  safe  to  say  that  we  must  accept  at  least 
two  great  periods,  the  older  Nyaya  and  Buddhism  up  to  the  10th 
century  of  our  era,  and  the  Navya  (new)  Nyaya  from  the  12th 
century  onwards. 

*  I  am  grateful  to  Prof.  M.  van  den  Oudenrijn  for  having  drawn  my  attention 
to  this  fact. 



Logic  shows  no  linear  continuity  of  evolution.  Its  history  resembles 
rather  a  broken  line.  From  modest  beginnings  it  usually  raises 
itself  to  a  notable  height  very  quickly  —  within  about  a  century  — 
but  then  the  decline  follows  as  fast.  Former  gains  are  forgotten,  the 
problems  are  no  longer  found  interesting,  or  the  very  possibility 
of  carrying  on  the  study  is  destroyed  by  political  and  cultural 
events.  Then,  after  centuries,  the  search  begins  anew.  Nothing 
of  the  old  wealth  remains  but  a  few  fragments;  building  on  those, 
logic  rises  again. 

We  might  therefore  suppose  that  the  evolution  of  logic  could  be 
presented  as  a  sine-curve;  a  long  decline  following  on  short  periods 
of  elevation.  But  such  a  picture  would  not  be  exact.  The  'new'  logic 
which  follows  on  a  period  of  barbarism  is  not  a  simple  expansion  of 
the  old ;  it  has  for  the  most  part  different  presuppositions  and  points 
of  view,  uses  a  different  technique  and  evolves  aspects  of  the 
problematic  that  previously  received  little  notice.  It  takes  on  a 
different  shape  from  the  logic  of  the  past. 

That  holds  in  the  temporal  dimension  for  western  and,  with  some 
limitations,  for  Indian  logic.  Perhaps  it  also  holds  in  the  spatial 
dimension  for  the  relation  between  the  two  considered  as  wholes. 
We  can  indeed  aptly  compare  Indian  logic  with  ancient  and  Scholas- 
tic logic  in  Europe,  as  lacking  the  notion  of  calculation;  but  beyond 
that  there  is  hardly  any  resemblance.  They  are  different  varieties 
of  logic.  It  is  difficult  to  fit  the  Indian  achievements  into  a  scheme 
of  evolution  in  the  west. 

The  essential  feature  of  the  whole  history  of  logic  seems  then  to  be 
the  appearance  of  different  varieties  of  this  science  separated  both 
in  time  and  space. 


There  are  in  essence,  so  far  as  we  can  determine,  four  such  forms: 

1.  The  Ancient  Variety  of  Logic.  In  this  period  logical  theorems 
are  mostly  formulated  in  the  object-language,  and  semantics  is  in 
being,  though  undeveloped.  The  logical  formulae  consist  of  words 
of  ordinary  language  with  addition  of  variables.  But  this  ordinary 
language  is  as  it  were  simplified,  in  that  the  chief  words  in  it  occur 
only  in  their  immediate  semantic  function.  The  basis  of  this  logic 
is  the  thought  as  expressed  in  natural  language,  and  the  syntactical 
laws  of  the  language  are  presupposed.  It  is  from  this  material  that 
the  ancient  logicians  abstract  their  formal  laws  and  rules. 

2.  The  Scholastic  Variety  of  Logic.  The  Scholastics  began  by  linking 
themselves  to  antiquity,  and  thus  far  simply  took  over  and  developed 
what  was  old.  But  from  the  end  of  the  12th  century  they  started  to 
construct   something   entirely   new.   This   logic  which   is   properly 



their  own  is  almost  all  formulated  metalogically.  It  is  based  on  and 
accompanied  by  an  accurate  and  well-developed  semantics.  For- 
mulae consist  of  words  from  ordinary  language,  with  very  few  or  no 
variables,  but  there  results  no  narrowing  of  the  semantic  functions 
as  in  antiquity.  Scholastic  logic  is  accordingly  a  thorough-going 
attempt  to  grasp  formal  laws  expressed  in  natural  language  (Latin 
with  plentifully  differentiated  syntactical  rules  and  semantic 
functions.  As  in  ancient  logic,  so  here  too  we  have  to  do  with 
abstraction  from  ordinary  language. 

3.  The  Mathematical  Variety  of  Logic.  Here  we  find  a  certain 
regress  to  the  ancient  variety.  Till  a  fairly  late  date  (about  1930; 
mathematical  logic  is  formulated  purely  in  the  object-language, 
with  rich  use  of  variables;  the  words  and  signs  used  have  narrowly 
limited  semantic  functions;  semantics  remains  almost  unnoticed 
and  plays  not  nearly  so  marked  a  role  as  in  the  Middle  Ages  and 
after  its  resurgence  since  about  1930.  Mathematical  logic  introduces 
two  novelties;  first,  the  use  of  an  artifical  language;  second,  and 
more  important,  the  constructive  development  of  logic.  This  last 
means  that  the  system  is  first  developed  formalistically  and  only 
afterwards  interpreted,  at  least  in  principle. 

Common  to  the  three  western  varieties  of  logic  is  a  far-reaching 
formalism  and  preponderantly  extensional  treatment  of  logical  laws. 

4.  The  Indian  Variety  of  Logic.  This  differs  from  the  western  in 
both  the  characteristics  just  mentioned.  Indian  logic  succeeds  is 
stating  certain  formal  laws,  but  formalism  is  little  developed  and  is 
obviously  considered  to  be  subsidiary.  Again  the  standpoint  is 
preponderantly  intensional  in  so  far  as  the  Indian  logicians  of  the 
last  period  knew  how  to  formulate  a  highly  developed  logic  of 
terms  without  employing  quantifiers. 

The  fore-going  arrangement  is  schematic  and  oversimplified, 
especially  in  regard  to  ancient  and  Indian  logic.  One  could  ask,  for 
instance,  whether  Megarian-Stoic  logic  really  belongs  to  the  same 
variety  as  Aristotelian,  or  whether  it  is  on  the  contrary  mainly 
new,  having  regard  to  its  markedly  semantic  attitude. 

Still  more  justified  would  perhaps  be  the  division  of  Indian  logic 
into  different  forms.  One  could  find,  for  example,  considerable 
justification  for  saying  that  Buddhist  logic  differs  notably  from  the 
strict  Nyaya  tradition  not  only  in  its  philosophical  basis,  nor  only 
in  details,  but  with  this  big  difference  that  the  Buddhists  show  a 
manifestly  extensional  tendency  in  contrast  to  the  Nyaya  com- 
mentators. Again,  evidence  is  not  lacking  that  the  Navy  a  Nyaya 
does  not  properly  exhibit  a  quite  new  type  of  logic,  since  in  some 
doctrines,  as  in  the  matter  of  Vyapti,  it  takes  over  Buddhist  modes 
of  expression,  in  others  it  follows  the  Nyaya  tradition,  in  others 
again  it  develops  a  new  set  of  problems  and  takes  up  a  fresh  stand- 



However  the  difference  between  Aristotle  and  the  Megarian- 
Stoic  school  seems  hardly  significant  enough  to  justify  speaking  of 
two  different  forms  of  logic.  As  to  Indian  logic  our  knowledge  is  so 
incomplete  that  it  would  be  rash  to  draft  a  division  and  charac- 
terization of  its  different  forms. 

A  further  problem  that  belongs  here  is  that  of  the  so-called 
'classical'  logic.  One  could  understand  it  as  a  distinct  variety,  since 
while  it  consists  of  fragments  of  scholastic  logic  (taking  over  for 
example  the  mnemonic  Barbara,  Celarenl  etc.,  yet  these  fragments 
are  interpreted  quite  unscholastically,  in  an  ancient  rather  than 
scholastic  way.  But  the  content  of  this  logic  is  so  poor,  it  is  loaded 
with  so  many  utter  misunderstandings,  and  its  creative  power  is  so 
extremely  weak,  that  one  can  hardly  risk  calling  something  so 
decadent  a  distinct  variety  of  logic  and  so  setting  it  on  a  level  with 
ancient,  scholastic,  mathematical  and  Indian  logic. 


We  said  above  that  every  new  variety  of  logic  contains  new 
logical  problems.  It  is  easy  to  find  examples  of  that:  in  Scholasticism 
there  are  the  magnificent  semiotic  investigations  about  the  proprie- 
taies  terminorum,  then  the  analysis  of  propositions  containing  time- 
variables,  investigations  about  quantifiers,  etc;  in  mathematical 
logic  the  problems  of  multiple  quantification,  description,  logical 
paradoxes,  and  so  on.  It  is  evident  that  quite  different  systems  of 
formal  logic  are  developed  as  a  result.  To  be  sure,  that  also  sometimes 
happens  within  the  framework  of  a  single  form  of  logic,  as  when  we 
single  out  Theophrastan  modal  logic  as  different  from  Aristotle's. 
The  class  of  alternative  systems  of  formal  logic  has  increased 
greatly  especially  since  Principia  Mathematica. 

One  might  therefore  get  the  impression  that  the  history  of  logic 
evidences  a  relativism  in  logical  doctrine,  i.e.  that  we  see  the  rise 
of  different  logics.  But  we  have  spoken  not  of  different  logics,  rather 
of  different  varieties  of  one  logic.  This  way  of  speaking  has  been 
chosen  for  speculative  reasons,  viz.  that  the  existence  of  many 
systems  of  logic  provides  no  proof  that  logic  is  relative.  There  is, 
further,  an  empirical  basis  for  speaking  of  one  logic.  For  history 
shows  us  not  only  the  emergence  of  new  problems  and  laws  but 
also,  and  perhaps  much  more  strikingly,  the  persistent  recurrence 
of  the  same  set  of  logical  problems. 

The  following  examples  may  serve  to  support  this  thesis: 

1.  The  problem  of  implication.  Posed  by  the  Megarians  and  Stoics 
(20.05  ff.),  it  was  resumed  by  the  Scholastics  (30.09  ff.),  and  again  by 
the  mathematical  logicians  (41.11  ff.).  Closely  connected  with  it, 
so  it  seems,  was  what  the  Indians  called  vyapli  (53.20,  54.07 f.). 
Perhaps  more  remarkable  is  the  fact  that  the  same  results  were 



reached  quite  independently  in  different  periods.  Thus  material 
implication  is  defined  in  just  the  same  way  by  Philo  (20.07),  Burleigh 
(30.14)  and  Peirce  (41.12 f.),  in  each  case  by  means  of  truth-values. 
Another  definition  is  also  first  found  among  the  Megarians  (20.10j, 
again,  and  this  time  as  their  main  concept  of  implication,  among  the 
Scholastics  (30.11  f.),  and  is  re-introduced  by  Lewis  in  1918  (49.04). 

2.  The  semantic  paradoxes  serve  as  a  second  example.  Already 
posed  in  the  time  of  Aristotle  (23.18),  discussed  by  the  Stoics 
(23.20),  the  problem  of  these  is  found  again  in  the  Scholastics 
(35.05 ff.),  and  forms  one  of  the  main  themes  in  mathematical 
logic  (§  48).  Re-discovery  of  the  same  solutions  is  again  in  evidence 
here,  e.g.  Russell's  vicious-circle  principle  was  already  known  to 
Paul  of  Venice. 

3.  A  third  group  of  problems  common  to  western  logic  is  that  of 
questions  about  modal  logic.  Posed  by  Aristotle  (§  15),  these  ques- 
tions were  thoroughly  gone  into  by  the  Scholastics  (§  33)  and  have 
taken  on  a  new  lease  of  life  in  the  latest  phase  of  mathematical 
logic  (49.03). 

4.  We  may  refer  again  to  the  analysis  of  quantifiers :  the  results  of 
Albert  of  Saxony  and  Peirce  are  based  on  the  same  understanding 
of  the  problem  and  run  exactly  parallel. 

5.  Similar  correspondences  can  be  noticed  between  Indian  and 
western  logic.  D.  H.  H.  Ingalls  has  recently  discovered  a  long 
series  of  problems  and  solutions  common  to  the  two  regions.  Most 
remarkable  is  the  fact  that  Indian  logic,  evolving  in  quite  different 
conditions  from  western,  and  independently  of  it,  eventually  dis- 
covered precisely  the  scholastic  syllogism,  and,  as  did  western  logic, 
made  its  central  problem  the  question  of  'necessary  connection'. 

Still  further  examples  could  be  adduced  in  this  connection;  it 
seems  as  though  there  is  in  the  history  of  logic  a  set  of  basic  problems, 
taken  up  again  and  again  in  spite  of  all  differences  of  standpoint,  and, 
still  more  important,  similarly  solved  again  and  again. 

Itis  not  too  easy  to  express  exactly,  but  every  reader  who  is  a 
logician  will  see  unmistakeably  the  community  of  mind,  by  which 
we  mean  the  recurrent  interest  in  certain  matters,  the  way  and 
style  of  treating  them,  among  all  inquirers  in  the  field  of  what  we 
comprehend  within  the  different  forms  of  formal  logic.  Read  in 
conjunction  our  texts  16.19,  22.16fL,  31.22,  33.20,  41.11  ff.  There 
can  be  no  doubt  that  the  same  attitude  and  spirit  is  expressed  in 
them  all. 


Closely  connected  with  the  question  of  the  unity  of  logic  is  the 
difficult  problem  of  its  progress.  One  thing  is  certain:  that  this 
problem  cannot  be  solved  a  priori  by  blind  belief  in  the  continuous 
growth  to  perfection  of  human  knowledge,  but  only  on  the  basis  of 



a  thoroughly  empirical  inquiry  into  detail.  We  can  only  learn 
whether  logic  has  progressed  in  the  course  of  its  history  from  that 
history  itself.  We  cannot  discover  it  by  means  of  a  philosophic 

But  the  problem  is  not  easily  solvable  with  our  present  historical 
knowledge.  One  question  which  it  involves  seems  indeed  to  be  safely 
answerable,  but  the  requisites  for  dealing  with  others  are  still 

We  can  safely  state  the  following: 

1.  The  history  of  logic  shows,  as  has  already  been  remarked,  no 
linear  ascending  development.  Consequently  in  the  case  of  an  advance, 
it  can  only  take  place  firstly,  within  a  given  period  and  form  of 
logic,  and  secondly,  so  as  to  raise  the  later  forms  to  a  higher  level 
than  the  earlier. 

2.  Some  advance  within  single  periods  and  forms  of  logic  is 
readily  perceivable.  We  can  see  it  best  in  Indian,  but  also  in  Scho- 
lastic and  mathematical  logic.  Every  particular  of  these  periods 
affords  a  safe  criterion  of  progress;  each  of  them  has  its  essential 
problems,  and  by  comparing  their  formulation  and  solution  in 
different  logicians  of  the  same  period  we  can  easily  see  that  the 
later  writers  pose  the  questions  more  sharply,  apply  better  method 
to  their  solution,  know  more  laws  and  rules. 

3.  If  the  history  of  logic  is  considered  as  a  whole,  here  too  a  certain 
advance  can  be  established  with  safety.  This  consists  in  the  fact 
that  new  problems  are  forthcoming  in  the  later  forms  of  logic. 
Thus  for  example  the  highly  wrought  semiotic  problematic  of  the 
Scholastics  is  quite  new  in  comparison  with  that  of  antiquity,  and 
therefore  also  more  complete;  the  logical  paradoxes  (not  the 
semantic  ones)  of  the  mathematical  logicians  are  new;  so  too  Albert 
of  Saxony's  problem  of  defining  quantifiers  is  new.  These  are  again 
only  some  examples  from  the  many  possible  ones. 

On  the  other  hand,  the  following  question  seems  to  be  still 
undecidable  in  the  present  state  of  knowledge:  taking  logic  as  a 
whole,  is  every  later  form  superior  to  all  earlier  ones  ? 

Too  often  this  question  is  answered  affirmatively  with  an  eye 
on  mathematical  logic,  particularly  because  people  compare  it  with 
its  immediate  predecessor,  'classical'  logic,  and  are  struck  by  the 
mass  of  laws  and  rules  which  calculation  makes  available  in  the 
new  form. 

But  'classical'  logic  is  by  no  means  to  be  equated  with  the  whole 
of  older  logic;  it  is  rather  a  decadent  form  of  our  science,  a  'dead 
period'  in  its  evolution.  Calculation,  again,  is  certainly  a  useful 
tool  for  logic,  but  only  as  facilitating  new  insights  into  logical 
interconnection.  It  is  undeniable  that  such  insights,  e.g.  in  the  logic 
of  relations,  have  been  reached  by  its  means,  and  the  convenience 
and  accuracy  of  this  instrument  are  so  great  that  no  serious  logician 



can  now  dispense  with  it.  But  we  would  not  go  so  far  as  to  say  that 
calculation  has  at  every  point  allowed  mathematical  logic  to  surpass 
the  older  forms.  Think  for  example  of  two-valued  prepositional 
logic:  the  essentially  new  features  introduced  by  Principia  Mathe- 
matical are  quite  unimportant  when  we  compare  the  scholastic 

Once  again  the  matter  reduces  to  our  insufficient  knowledge  of 
the  earlier  forms  of  logic.  For  years  people  spoke  of  a  supposed 
great  discovery  by  De  Morgan;  then  Lukasiewicz  showed  that  his 
famous  law  was  part  of  the  elementary  doctrine  of  Scholasticism. 
The  discovery  of  truth-matrices  was  ascribed  to  Peirce,  or  even 
Wittgenstein;  Peirce  himself  found  it  in  the  Megarians.  D.  Ingalls 
found  Frege's  classical  definition  of  number  in  the  Indian  Mathu- 
ranatha  (17th  century).  And  then  we  are  all  too  well  aware  that  we 
know,  as  has  been  said,  only  fragments  of  Scholastic  and  Indian 
logic,  while  much  more  awaits  us  in  manuscripts  and  even  in  unread 
printed  works.  The  Megarian-Stoic  logic,  too,  is  lost,  except  for  a 
few  poor  fragments  transmitted  by  its  opponents. 

Also  highly  relevant  to  the  question  of  the  continual  progress  of 
logic  throughout  its  history  is  the  fact  that  the  earlier  varieties 
are  not  simply  predecessors  of  contemporary  logic,  but  deal  in  part 
with  the  same  or  similar  problems  though  from  a  different  stand- 
point and  by  different  methods.  Now  it  is  hard  for  a  logician  trained 
in  the  contemporary  variety  of  logic  to  think  himself  into  another. 
In  other  words,  it  is  hard  for  him  to  find  a  criterion  of  comparison. 
He  is  constantly  tempted  to  consider  valuable  only  what  fits  into 
the  categories  of  his  own  logic.  Impressed  by  our  technique,  which  is 
not  by  itself  properly  logic,  having  only  superficial  knowledge  of 
past  forms,  judging  from  a  particular  standpoint,  we  too  often  risk 
misunderstanding  and  under-rating  other  forms. 

Even  in  the  present  state  of  knowledge  we  can  be  sure  that 
various  points  about  the  older  forms  still  escape  our  comprehension. 
One  example  is  the  Scholastic  doctrine  of  supposition,  which  is 
evidently  richer  in  important  insights  and  rules  than  the  semiotic  so 
far  developed  by  mathematical  logic.  Another  is  perhaps  the 
treatment  of  implication  (vyapti)  by  the  thinkers  of  the  Navya 
Nyaya.  Still  further  examples  could  be  given. 

Again,  when  an  unprejudiced  logician  reads  some  late-Scholastic 
texts,  or  it  may  be  some  Stoic  fragments,  he  cannot  resist  the  im- 
pression that  their  general  logical  level,  their  freedom  of  movement 
in  a  very  abstract  domain,  their  exactness  of  formulation,  while  they 
are  equalled  in  our  time,  have  by  no  means  been  excelled.  The  modern 
mathematical  logician  certainly  has  a  strong  support  in  his  calculus, 
but  all  too  frequently  that  same  calculus  leads  him  to  dispense 
with  thought  just  where  it  may  be  most  required.  A  conspicuous 
example  of  this  danger  is  provided  by  statements  made  for  long 



years  by  mathematical  logicians  concerning  the  problem  of  the  null 

These  considerations  tell  against  the  thesis  that  logic  has  pro- 
gressed as  a  whole,  i.e.  from  variety  to  variety;  it  looks  as  though  we 
have  insufficient  grounds  for  holding  it.  But  of  course  it  does  not  at 
all  follow  that  another  thesis,  viz.  of  a  purely  cyclic  development  of 
formal  logic  with  continual  recurrence  of  the  same  culminating 
points,  is  sufficiently  established. 

The  historian  can  only  say;  we  do  not  know  whether  there  is  an 
over-all  progress  in  the  history  of  logic. 


Conformably  to  the  directions  of  the  series  Orbis  Academicus  this 
work  will  present  a  documented  history  of  problems. 

We  are  not,  therefore,  presenting  a  material  history  of  logic 
dealing  with  everything  that  has  any  historical  importance,  but  a 
delineation  of  the  history  of  the  problematic  together  with  the 
complex  of  essential  ideas  and  methods  that  are  closely  connected 
with  it.  We  only  take  into  account  those  periods  which  have  made  an 
essential  contribution  to  the  problematic,  and  among  logicians  those 
who  seem  to  us  to  rank  as  specially  good  representatives  of  their 
period.  In  this  connection  some  thinkers  of  outstanding  importance, 
Aristotle  above  all,  Frege  too,  will  receive  much  fuller  treatment  than 
would  be  permissible  in  a  material  history. 

The  story  will  be  told  with  the  help  of  texts,  and  those  originally 
written  in  a  foreign  language  have  been  translated  into  English. 
This  procedure,  unusual  in  a  scientific  work,  is  justified  by  the  consi- 
deration that  only  a  few  readers  could  understand  all  the  texts  if 
they  were  adduced  in  their  original  language.  For  even  those  readers 
with  some  competence  in  Greek  are  not  automatically  able  to  under- 
stand with  ease  a  text  of  formal  logic  in  that  tongue.  But  the  spe- 
cialist logician  will  easiliy  be  able  to  find  the  original  text  by  reference 
to  the  sources. 

The  passages  quoted  will  be  fairly  thoroughly  commented  where 
this  seems  useful,  for  without  some  commentary  many  of  them 
would  not  be  readily  intelligible. 


In  itself  such  a  history  admits  of  being  arranged  according  to 
problems.  One  could  consider  first  questions  of  semiotics,  then 
propositional  ones,  then  those  of  predicate  logic  etc.,  so  as  to  pursue 
the  whole  history  of  each  class  of  problems.  E.g.  the  chapter  on 
propositional  logic  could  begin  with  Aristotle,  go  on  to  the  Megarian- 



Stoic  theory  of  Xoyo<;,  then  to  the  scholastic  consequentiae,  to  the 
propositional  interpretation  of  the  Boolean  calculus,  to  McColl,  Peirce 

and  Frege,  to  chapters  2 — 5  of  the  Principia,  finally  to  Lukasiewicz. 

Such  a  method  of  treatment  is,  however,  forbidden  by  the  non- 
linear evolution  of  logic,  and  above  all  by  the  fact  that  it  takes  on  a 
different  form  in  every  epoch.  For  every  particular  group  of  pro- 
blems within  one  variety  is  closely  connected  with  other  complexes 
of  problems  in  the  same  variety.  Torn  from  its  context  and  ranged 
with  the  cognate  problems  in  another  variety,  it  would  be,  not  just 
unintelligible,  but  quite  misunderstood.  The  problem  of  impli- 
cation provides  a  good  example:  the  Scholastics  put  it  in  the 
context  of  their  theory  of  meaning,  and  their  theory  is  not  to  In- 
understood  apart  from  that.  Every  problem  considered  in  a  given 
variety  of  logic  needs  viewing  in  the  context  of  the  total  problematic 
of  that  variety. 

It  is  necessary,  therefore,  to  arrange  the  whole  history  according 
to  the  varieties  of  logic.  Within  each  we  have  tried  to  show  the 
connection  of  the  various  groups  of  problems.  This  has,  however,  not 
proved  to  be  the  best  course  everywhere.  In  the  discussion  of 
antiquity  a  grouping  of  the  material  according  to  the  chronology  of 
logicians  and  schools  has  seemed  preferable,  especially  because  one 
logician,  Aristotle,  has  an  incomparably  great  importance. 


As  our  knowledge  of  many  domains  is  still  very  fragmentary,  we 
cannot  aim  at  completeness.  One  period  that  is  probably  fairly 
important,  the  Arabian,  cannot  be  noticed  at  all.  Citations  from 
Scholasticism  are  certainly  only  fragments.  Even  our  knowledge  of 
ancient  and  mathematical  logic  is  far  from  satisfactory.  Conse- 
quently this  work  serves  rather  as  a  survey  of  some  aspects  of  the 
history  of  logical  problems  than  as  a  compilation  of  all  that  is  essen- 
tial to  it. 

What  is  rather  aimed  at  is  a  general  orientation  in  whatever  kind 
of  problems,  methods  and  notions  is  proper  to  each  variety  of  logic, 
and  by  that  means  some  presentation  of  the  general  course  of  the 
history  of  logic  and  its  laws.  The  emphasis  will  be  put  on  this  course 
of  the  problematic  as  a  whole. 

Hence  we  have  also  decided  to  risk  a  short  account  of  Indian 
logic,  in  spite  of  subjective  and  objective  reasons  to  the  contrary. 
For  this  logic  seems  to  be  of  great  interest  precisely  with  reference  to 
the  laws  of  the  whole  development.  At  the  same  time  it  is  the  only 
form  which  has  developed  quite  independently  of  the  others.  The 
chapter  on  Indian  logic  must,  however,  be  managed  differently 
from  the  rest,  not  only  because  our  knowledge  of  the  subject  is  even 
less  sufficient  than  of  Scholastic  logic,  but  also  because  we  have  to 
rely  on  translations.  This  chapter  will  be  treated  as  a  kind  of  appendix. 




In  order  to  establish  a  comparison  between  the  problems  and 
theorems  which  have  been  formulated  in  different  epochs  and 
languages,  we  have  had  to  use  a  unified  terminology  in  our  comments. 
For  the  most  part  we  have  taken  this  from  the  vocabulary  of 
contemporary  formal  logic.  But  as  this  vocabulary  is  not  at  all 
familiar  to  the  majority  of  readers,  we  shall  here  explain  the  most 
important  technical  expressions. 


By  'expression',  'formula',  'word',  'symbol'  etc.  we  here  intend 
what  Morris  calls  the  sign-vehicle,  and  so  the  material  component 
of  the  sign ;  i.e.  a  certain  quantity  of  ink,  or  bundle  of  sound  waves.  A 
specially  important  class  of  expression  is  that  of  sentences,  i.e. 
expressions  which  can  be  characterized  as  true  or  false.  It  must  be 
stressed  that  a  sentence,  so  understood,  is  an  expression,  a  material 
sign,  and  not  what  that  signs  stands  for.  The  word  'proposition'  has 
been  variously  used,  as  synonymous  with  'sentence'  in  the  sense 
just  explained  (cf.  26.03),  more  normally  for  a  sentence  precisely  as 
meaningful  (Scholastics  generally),  sometimes  with  various  psycho- 
logical and  subjective  connotations  (cf.  the  'judgment'  of  the 
'classical'  logicians),  nowadays  commonly  as  the  objective  content  of 
a  meaningful  sentence  (cf.  the  Stoic  a^icopia).  In  our  commentaries 
we  keep  'sentence'  for  the  material  expression,  as  above  and  use 
'proposition'  in  the  sense  appropriate  to  the  historical  context  and  as 
indicated  by  normal  usage,  which  seems  frequently  to  approximate 
to  that  of  the  Scholastics. 

We  divide  expressions  into  atomic  and  molecular  (the  thought  is 
Aristotelian,  cf.  10.14  and  10.24),  the  former  being  without  parts 
that  are  themselves  expressions  of  the  given  language,  the  latter  con- 
taining such  parts.  Molecular  expressions  are  analysed  sometimes 
into  subject  and  predicate  in  accordance  with  the  tradition  of  Ari- 
stotle and  the  Scholastics,  sometimes  into  functor  and  argument. 
The  functor  is  the  determining  element,  the  argument  the  one  deter- 
mined ;  this  is  also  true  of  predicate  and  subject  respectively,  but  the 
other  pair  of  terms  is  more  general  in  applicability.  'And',  'not', 
names  of  relations,  are  thought  of  as  functors. 

We  distinguish  between  constant  and  variable  expressions  (again 
with  Aristotle,  cf.  13.04),  called  constants  and  variables  for  short.  The 
former  have  a  determinate  sense,  the  latter  only  serve  to  mark  void 
places  in  which  constants  can  be  substituted.  Thus,  for  example,  in 
'x  smokes',  V  is  a  variable  and  'smokes'  a  constant.  With  Frege 
(42.02)  we  call  a  molecular  expression  which  exhibits  a  variable  a 
function.  Thus  we  speak  of  propositional  functions,  that  is  to  say  of 



expressions  which,  if  the  variables  that  occur  in  them  are  properly 
replaced  by  constants,  become  sentences  (or  propositions  in  the 
Scholastic  sense).  lx  smokes'  is  such  a  propositional  function. 

Among  propositional  functions  we  often  mention  the  logical 
sum  or  inclusive  disjunction  of  two  propositions  or  terms,  the  logical 
product  or  conjunction,  implication  and  equivalence.  Quantifiers  (cf. 
44.01),  'all',  'some',  'for  every  x\  'there  is  a  y  such  that',  are  some- 
times counted  as  functors. 

Variables  which  can  only  be  meaningfully  replaced  by  propositions 
we  call  propositional  variables;  such  as  can  only  be  meaningfully 
replaced  by  terms  we  call  term-variables.  Correspondingly  we  speak 
of  laws  of  propositional  logic  and  term-logic.  Term-logic  is  divided 
into  predicate-,  class-  (or  classial),  and  relation-logic.  Predicate- 
logic  treats  of  intensions,  class-logic  of  extensions;  relation-logic  is 
the  theory  of  those  special  formal  properties  which  belong  to  rela- 
tions, e.g.  symmetry  (if  Ft  holds  between  a  and  6,  then  it  also  holds 
between  b  and  a),  transitivity  (if  B  holds  between  a  and  b  and  be- 
tween b  and  c,  then  it  also  holds  between  a  and  c)  etc. 

The  general  doctrine  of  signs  we  call,  with  W.  Morris  (5.01), 
semioiic.  This  is  divided  into  syntax  (theory  of  the  relationships 
between  signs),  semantics  (theory  of  the  relationships  between  signs 
and  their  significates),  and  pragmatics  (theory  of  the  relationships 
between  signs  and  those  who  use  them).  Correspondingly  we  speak 
of  syntactical,  semantic  and  pragmatic  laws  and  theories.  In  the 
field  of  semantics  we  distinguish  between  the  denotation  and  the 
meaning  or  sense  of  a  sign  —  which  denotes  the  object  of  which  it 
is  a  sign  (its  reference),  and  means  its  content.  (In  translating 
Scholastic  texts  we  use  'signifies'  for  'significaV  and  leave  further 
determination  to  be  judged,  where  possible,  by  the  context.)  Thus 
for  example,  the  word  'horse'  denotes  a  horse,  but  means  what 
makes  a  horse  a  horse,  what  we  might  call  'horseness'.  We  dis- 
tinguish further  between  object-language  in  which  the  signs  denote 
objects  that  are  not  part  of  the  language,  and  the  corresponding 
meta-language  in  which  the  signs  denote  those  of  the  object-language. 
In  accordance  with  this  terminology  the  word  'cat'  in  the  sentence 
'a  cat  is  an  animal'  belongs  to  the  object-language  since  it  denotes 
a  non-linguistic  object,  but  in  the  sentence  '"cat"  is  a  substantive' 
it  belongs  to  the  meta-language,  since  it  denotes  the  word  'cat'  and 
not  a  cat  itself.  When  an  expression  is  used  as  the  name  of  another 
expression  that  has  the  same  form,  we  follow  the  prescription  of 
Frege  (39.03)  and  write  it  between  quotation-marks. 

Finally  we  distinguish  between  logical  laws  and  rules,  as  did  the 
Stoics  (§  22,  A  and  B)  and  Scholastics  (cf.  the  commentary  on  31.13). 
Laws  state  what  is  the  case,  rules  authorize  one  to  proceed  in  such 
and  such  a  way. 




In  divergence  from  the  widespread  practice,  which  is  that  of  the 
author  himself,  all  use  of  mathematico-logical  symbolism  has  been 
avoided  in  the  commentaries  on  texts  not  of  this  character.  In  many 
cases  this  symbolism  affords  easy  abbreviation,  and  laws  formulated 
by  its  means  are  much  easier  for  the  specialist  to  read  than  verbally 
expressed  propositional  functions  or  propositions.  But  two  reasons 
militate  against  its  use: 

1.  First,  objectively,  it  introduces  an  appreciable  risk  of  misunder- 
standing the  text.  Such  a  risk  is  present  in  every  case  of  translation, 
but  it  is  particularly  great  when  one  uses  a  terminology  with  so 
narrowly  defined  a  sense  as  that  of  mathematical  logic.  Take,  for 
example,  signs  of  implication.  Those  at  our  disposal  essentially 
reduce  to  two:  'd'  and  'F'.  Which  of  them  are  we  to  use  to  express 
Diodorean  implication?  Certainly  not  the  first,  for  that  means 
Philonian  implication;  but  not  the  second  either,  for  that  would 
mean  that  one  was  sure  that  Diodorus  defined  implication  just  like 
Lewis  or  Buridan,  which  is  by  no  means  certain.  Another  example 
is  the  Peano-Russellian  paraphrase  of  Aristotelian  syllogistic  as  it 
occurs  in  the  Principia  (5.02).  It  is  undoubtedly  a  misinterpretation 
of  Aristotle's  thought,  for  it  falsifies  many  laws  of  the  syllogistic 
which  on  another  interpretation  (that  of  Lukasiewicz)  can  be  seen 
to  be  correct  (5.03). 

Some  notions  not  deriving  from  mathematical  logic  could  indeed 
be  expressed  in  its  symbolism,  e.g.  the  Philonian  implication  or  that 
of  Buridan;  but  to  single  out  these  for  such  interpretation  and  to 
make  use  of  verbal  formulation  in  other  cases  would  be  to  cause  a 
complication  that  is  better  avoided. 

But  of  course  that  is  not  to  say  that  no  such  symbolism  ought 
to  be  employed  for  any  form  of  logic.  For  particular  logicians,  or 
a  particular  form,  the  use  of  an  artificial  symbolism  is  not  only 
possible,  but  to  be  desired.  But  then  every  case  requires  a  special 
symbolism.  What  we  cannot  do  is  to  create  a  unique  symbolism 
suitable  for  all  the  ideas  that  have  been  developed  in  the  different 
varieties  of  logic. 

2.  A  subjective  reason  is  provided  by  the  limits  of  the  work, 
which  aims  to  make  allowance  for  the  reader  who  is  formed  in  the 
humanities  but  innocent  of  mathematics.  For  such,  and  they  are 
obviously  the  majority,  mathematico-logical  symbolism  would 
not  clarify  his  reading,  but  cloud  it  unnecessarily. 

In  these  circumstances  we  have  been  at  pains  to  use  such  texts 
as  exhibit  no  artificial  symbolism,  even  in  the  chapter  on  mathe- 
matical logic,  so  far  as  that  is  possible.  Symbolic  texts  are  of  course 
cited  as  well,  and  in  such  a  way  that  one  who  wishes  to  acquire  the 
symbolic  language  of  mathematical  logic  can  learn  the  essentials 
from  this  work.  But  the  texts  which  treat  of  the  basic  problems  of 



logic  have  been  chosen  in  such  a  way  that  they  are  as  far  as  possible 
intelligible  without  a  knowledge  of  this  symbolism. 


All  texts  are  numbered  decimally,  the  integral  part  giving  the 
paragraph  in  which  the  citation  occurs,  the  decimals  referring  to  a 
consecutive  numbering  within  the  paragraph. 

Texts  are  set  in  larger  type  than  the  commentaries,  except  for 
formulas  due  to  the  author,  which  are  also  in  larger  type. 

Added  words  are  enclosed  in  round  parentheses.  Expressions  in 
square  parentheses  occur  thus  in  the  text  itself.  Formulas  are  an 
exception  to  this:  all  parentheses  occurring  in  them,  together  with 
their  contents,  occur  so  in  the  original  texts. 

Quotation  marks  and  italics  in  ancient  and  scholastic  texts  are 
due  to  the  author. 

Remarks  concerning  textual  criticism  are  presented  in  starred 

Special  points  concerning  the  chapter  on  Indian  logic  are  stated 
in  §  50,  D. 



The  Greek  Variety  of  Logic 



Aristotle,  the  first  historian  of  philosophy,  calls  Zeno  of  Elea  the 
'founder  of  dialectic'  (6.01),  but  the  first  two  men,  so  far  as  we  know, 
to  reflect  seriously  on  logical  problems  were  Plato  and  Euclid  of 
Megara,  both  pupils  of  Socrates.  And  as  Aristotle  himself  ascribes 
to  Socrates  important  services  in  the  domain  of  logic  (6.02),  or 
rather  of  methodology  from  which  logic  later  developed,  perhaps 
Socrates  should  be  considered  to  be  the  father  of  Greek  logic. 

Aristotle  was  a  pupil  of  Plato,  and  his  logic  undoubtedly  grew  out 
of  the  practice  of  the  Platonic  Academy.  Aristotle's  chief  pupil 
and  long-time  collaborator,  Theophrastus,  provides  the  link  between 
the  logical  thought  of  his  master  and  that  of  the  Stoa.  For  con- 
temporaneously and  parallel  with  Aristotelian  logic  there  developed 
that  derived  from  Euclid,  of  which  the  first  important  representatives 
were  Megarians,  Diodorus  Cronus,  Philo  of  Megara  and  others; 
later  came  the  Stoics,  who  were  closely  connected  with  the  Megarians, 
having  Chrysippus  as  their  most  important  thinker. 

After  the  death  of  Chrysippus,  disputes  arose  between  the  Peripa- 
tetic and  Megarian-Stoic  schools,  the  latter  now  represented  by  the 
Stoics  alone,  and  syncretism  became  prominent.  Even  then  logicians 
were  not  lacking,  the  more  important  among  them  being  apparently 
the  commentators  on  Aristotle's  logical  works  (Alexander,  Philo- 
ponus),  many  Sceptics  (especially  Sextus  Empiricus),  these  in  the 
3rd  century  B.C.,  and  finally  Boethius  (5th-6th  century  a.d.). 

The  following  table  shows  the  chronological  and  doctrinal 
connection  down  to  Chrysippus: 

Zeno  of  Elea,  c.  464/60  b.c. 

Socrates  ob.  399  The  ancient  Sophists 

Plato  428/7-348/47  Euclid  of  Megara  c.  400 

Aristotle  384-322 

Theophrastus  ob.  287/86 

■Diodorus  Cronus  ob.  307 

rJL/lUUUl  US  Kj 
Philo  o 

f  Megara 

*  Zeno  of  Citium 



Chrysippus  of  Soli 

Peripatetic  School  Stoic  School         Megarian  School 




The  problematic  of  formal  logic  by  and  large  began  with  Aristotle. 
He  was  undoubtedly  the  most  fertile  logician  there  has  ever  been, 
in  the  sense  that  a  great  many  logical  problems  were  raised  for  the 
first  time  in  his  works.  Close  to  him  in  the  history  of  ancient  logic 
is  a  group  of  thinkers  who  are  nearly  as  important,  the  Megarian- 
Stoic  school.  Aristotle  lived  in  the  4th  century  B.C.;  the  essential 
development  of  the  Megarian-Stoic  school  can  be  thought  of  as 
ending  with  the  death  of  Ghrysippus  of  Soli  at  the  end  of  the  3rd 
century  B.C.  Hence  in  Greek  antiquity  there  is  a  relatively  short 
period  to  be  considered,  from  the  second  half  of  the  4th  to  the  end 
of  the  3rd  century  B.C. 

But  that  does  not  mean  that  there  was  no  logical  problematic 
outside  those  150  years.  Even  before  Aristotle,  a  problematic 
emerged  in  the  form  of  the  pre-Socratic  and  Platonic  dialectic, 
admittedly  without  ever  developing  into  a  logical  theory.  Again, 
long  after  the  death  of  Chrysippus,  and  right  on  to  the  end  of  anti- 
quity, i.e.  to  the  death  of  Boethius  (6th  century  a.d.),  many  reflec- 
tions on  logical  problems  are  to  be  found  in  the  so-called  Commen- 
tators. This  last  period  is  not  comparable  in  fruitfulness  with  that  of 
Aristotle  and  the  Stoics,  but  we  are  indebted  to  it  for  various 
insights  worth  remark. 

Accordingly,  from  our  point  of  view,  antiquity  is  divided  into 
three  main  periods: 

1.  the  preparatory  period,  to  the  time  when  Aristotle  began  to 
edit  his  Topics. 

2.  the  Aristotelian-Megarian-Stoic  period,  occupying  the  second 
half  of  the  4th  to  the  end  of  the  3rd  century  B.C. 

3.  the  period  of  the  Commentators,  from  about  200  b.c.  to  the 
death  of  Boethius  at  the  beginning  of  the  6th.  century  a.d. 

The  second  of  these  periods  is  so  outstandingly  important  that 
it  is  appropriate  to  divide  it  into  two  sections  covering  respectively 
Aristotle  and  the  Megarian-Stoic  school.  We  have  then  four  tempo- 
rally distinct  sections:  1.  pre-Aristotelians,  2.  Aristotle  and  his 
immediate  pupils,  3.  the  Megarian-Stoic  school,  4.  Commentators. 


The  history  of  Greek  logic  is  the  relatively  best-known  period  in 
the  development  of  formal  logic.  By  contrast  with  the  Middle 
Ages  and  after,  and  to  some  extent  with  logistic  too,  nearly  all  the 
surviving  texts  of  the  logicians  of  this  age  are  readily  available  in 
good  modern  editions,  together  with  a  whole  series  of  scientific 
treatises  on  their  contents.  In  this  connection  there  are  two  classes 
of  works: 



a)  On  the  one  hand  the  philologists  have  been  busy  for  more 
than  a  century  with  solving  numerous  and  often  difficult  problems 
of  literary  history  relevant  to  ancient  logic.  Yet  great  as  is  the  debt 
of  gratitude  owed  by  logicians  to  this  immense  work,  one  cannot 
pass  over  the  fact  that  most  philologists  lack  training  in  formal  logic 
and  so  too  often  overlook  just  the  most  interesting  of  the  ancient 
texts.  Mostly,  too,  their  interest  centres  on  ontological,  metaphysical, 
epistemological  and  psychological  questions,  so  that  logic  comes  to 
be  almost  always  neglected.  To  quote  only  one  example:  logic  is 
allotted  few  pages  in  Polenz's  two  big  volumes  on  the  Stoa.  Then 
again  editions  made  without  a  thorough  logical  training  are  often 
insufficient:  Kochalsky's  edition  of  Stoic  fragments  may  serve  as 
an  instance. 

b)  On  the  other  hand  logicians  too,  especially  since  the  pioneer 
work  of  G.  Vailati  (1904)  and  A.  Rustow  (1908),  have  considered  a 
fair  number  of  problems  arising  from  these  texts.  Epoch-making 
in  this  field  is  the  article  Zur  Geschichle  der  Aussagenlogik  (1935) 
by  J.  Lukasiewicz.  The  same  scholar  has  given  us  books  on  the 
principle  of  contradiction  in  Aristotle  and  the  Aristotelian  (assertoric) 
syllogistic.  Important  too  are  the  researches  of  H.  Scholz  whose 
Geschichle  der  Logik  appeared  in  1931  and  who  has  written  a  number 
of  other  studies.  Each  of  these  has  formed  a  small  school.  J.  Sala- 
mucha  investigated  the  concept  of  deduction  in  Aristotle  (1930). 
I.  M.  Bochenski  wrote  a  monograph  on  Theophrastus  (1939);  his 
pupils  J.  Stakelum  (1940)  and  R.  van  den  Driessche  (1948)  published 
studies  on  the  period  of  the  Commentators,  the  former  dealing  with 
Galen,  the  latter  with  Boethius.  Boethius  has  also  been  dealt  with 
by  K.  Diirr  (1952).  A.  Becker,  a  pupil  of  Scholz,  produced  an  impor- 
tant work  on  Aristotle's  modal  syllogisms  (1933).  B.  Mates,  influen- 
ced by  Lukasiewicz,  has  made  a  thorough  study  of  Stoic  logic  (1953). 

The  state  of  inquiry  up  to  now  may  be  characterized  thus: 
Aristotelian  studies  are  well  opened  up,  though  much  is  still  missing, 
e.g.  discussion  of  the  Topics;  good  editions  of  the  text  are  also 
available.  We  also  have  a  very  fair  knowledge  of  Megarian-Stoic 
logic,  though  fresh  editions  of  the  texts  are  desirable.  Very  little 
work  has  been  done  on  the  period  of  the  Commentators,  but  good 
editions  are  mostly  to  hand.  The  pre-Aristotelian  period  is  also  very 
insufficiently  explored,  notwithstanding  the  valuable  studies  by 
A.  Krokiewicz,  a  philologist  with  logical  training.  Especially 
desirable  is  a  thorough-going  treatment  of  the  beginnings  of  logic 
in  Plato,  though  admittedly  such  a  work  would  meet  with  con- 
siderable difficulties. 

More  exact  information  about  the  literature  will  be  found  in  the 




When  Aristotle  brought  to  a  close  the  earliest  part  of  his  logical 

work,  i.e.  the  Topics  and  De  Sophislicis  Elenchis,  he  could  proudly 
write : 

7.01  In  the  case  of  all  discoveries  the  results  of  previous 
labours  that  have  been  handed  down  from  others  have  been 
advanced  bit  by  bit  by  those  who  have  taken  them  on, 
whereas  the  original  discoveries  generally  make  an  advance 
that  is  small  at  first  though  much  more  useful  than  the  develop- 
ment which  later  springs  out  of  them.  For  it  may  be  that  in 
everything,  as  the  saying  is,  'the  first  start  is  the  main  part' : 
and  for  this  reason  also  it  is  the  most  difficult;  .  .  . 

Of  this  inquiry,  on  the  other  hand,  it  was  not  the  case  that 
part  of  the  work  had  been  thoroughly  done  before,  while  part 
had  not.  Nothing  existed  at  all. 


What  Aristotle  says  of  'this  inquiry'  of  his  seems  still  to  hold  good ; 
we  know  of  no  logic,  i.e.  an  elaborated  doctrine  of  rules  or  laws, 
earlier  than  the  Topics.  Certain  rules  of  inference,  however,  appear 
to  have  been  consciously  applied  long  before  Aristotle  by  many 
Greeks,  without  being  reflectively  formulated,  much  less  axiomatized. 
Aristotle  himself  says  elsewhere  that  Zeno  of  Elea  was  the  'founder 
of  dialectic'  (6.01),  and  it  is  in  fact  hardly  possible  that  Zeno 
formulated  his  famous  paradoxes  without  being  aware  of  the  rules 
he  was  applying.  The  texts  ascribed  to  him  are  only  to  be  found  in 
late  commentators,  including,  however,  Simplicius  who  was  a 
serious  investigator;  criticism  casts  no  doubt  on  their  authenticity. 
We  give  some  examples  of  his  dialectic : 

7.02  In  the  case  that  they  (beings)  are  many,  they  must 
be  as  many  as  they  are,  neither  more  nor  less.  But  if  they  are 
as  many  as  they  are,  then  they  are  limited  (determinate).  If 
(however)  beings  are  many,  then  they  are  unlimited  (indeter- 
minate) :  since  there  are  yet  other  beings  between  the  beings 
and  others  again  between  those.  And  thus  beings  are  unlimited 

7.03  If  beings  are,  every  one  must  have  magnitude  and  volume, 
and  one  part  of  it  must  be  distinct  from  another  .  .  .  And  so, 
if  they  are  many  they  must  be  at  once  small  and  great;  small, 



since  they  have  no  magnitude,  and  great  since  they  are  un- 
limited (indeterminate). 

7.04  If  there  is  a  place,  it  is  in  something;  for  every  being 
is  in  something;  but  what  is  in  something  is  also  in  a  place. 
Hence  the  place  will  itself  be  in  a  place,  and  so  on  without 
end;  hence  there  is  no  place. 

G.  Vailati  stressed  a  text  from  Plato  in  which  a  similar  process  of 
inference  is  used : 

7.05  Socrates:  And  the  best  of  the  joke  is,  that  he  acknow- 
ledges the  truth  of  their  opinion  who  believe  his  own  opinion 
to  be  false;  for  he  admits  that  the  opinions  of  all  men  are  true. 

Theodorus:  Certainly. 

Socrates:  And  does  he  not  allow  that  his  own  opinion  is 
false,  if  he  admits  that  the  opinion  of  those  who  think  him 
false  is  true  ? 

Theodorus :  Of  course. 

Socrates:  Whereas  the  other  side  do  not  admit  that  they 
speak  falsely? 

Theodorus:  They  do  not. 

Socrates:  And  he,  as  may  be  inferred  from  his  writings, 
agrees  that  this  opinion  is  also  true. 

Theodorus:  Clearly. 

Socrates:  Then  all  mankind,  beginning  with  Protagoras,  will 
contend,  or  rather,  I  should  say  that  he  will  allow,  when  he 
concedes  that  his  adversary  has  a  true  opinion,  Protagoras, 
I  say,  will  himself  allow  that  neither  a  dog  nor  any  ordinary 
man  is  the  measure  of  anything  which  he  has  not  learned  -  am 
I  not  right? 

Theodorus:  Yes. 

The  big  fragment  of  Gorgias  (7.06)  also  contains  something 
similar,  but  this  is  so  evidently  composed  in  the  technical  terminology 
of  the  Stoics  and  betrays  so  highly  developed  a  technique  of  logical 
thought  that  we  cannot  ascribe  it  to  the  Sophists,  nor  even  to 
Aristotle.  It  is,  however,  possible  that  the  young  Aristotle  did  indeed 
formulate  the  famous  proof  of  the  necessity  of  philosophy  in  the  way 
which  it  ascribed  to  him.  This  proof  is  transmitted  to  us  in  the  follow- 
ing three  passages  among  others: 

7.07  There  are  cases  in  which,  whatever  view  we  adopt,  we 
can  refute  on  that  ground  a  proposition  under  consideration. 
So  for  instance,  if  someone  was  to  say  that  it  is  needless  to 
philosophize :  since  the  enquiry  whether  one  needs  to  philo- 



sophize  or  not  involves  philosophizing,  as  he  (Aristotle)  has 
himself  said  in  the  Protreplicus,  and  since  the  exercise  of 
a  philosophical  pursuit  is  itself  to  philosophize.  In  showing 
that  both  positions  characterize  the  man  in  every  case,  we 
shall  refute  the  thesis  propounded.  In  this  case  one  can  rest 
one's  proof  on  both  views. 

7.08  Or  as  Aristotle  says  in  the  work  entitled  Protreplicus 
in  which  he  encourages  the  young  to  philosophize.   For  he 
says:  if  one  must  philosophize,  then  one  must  philosophize;  if 
one  does  not  have  to  philosophize,  one  must  still  philosophize. 
So  in  any  case  one  has  to  philosophize. 

7.09  Of  the  same  kind  is  the  Aristotelian  dictum  in  the 
Protreplicus:  whether  one  has  to  philosophize  or  not,  one 
must  philosophize.  But  either  one  must  philosophize  or  not; 
hence  one  must  in  any  case  philosophize. 


All  the  texts  adduced  above  spring  from  the  milieu  of  'dialectic'. 
This  word  that  is  later  given  so  many  meanings  and  is  so  mis-used 
originally  had  the  same  meaning  as  our  'discussion'.  It  is  a  matter 
of  disputation  between  two  speakers  or  writers.  That  is  probably  the 
reason  why  most  of  the  rules  of  inference  used  here  -  termed,  as  it 
seems,  iogoi'  -  lead  to  negative  conclusions:  the  purpose  was  to 
refute  something,  to  show  that  the  assertion  propounded  by  the 
opponent  is  false. 

This  suggests  the  conjecture  that  these  logoi  belong  to  the  field 
of  propositional  logic,  that  is  to  say  that  it  is  here  a  matter  of 
logical  relations  between  propositions  as  wholes  without  any  ana- 
lysis of  their  structure.  And  in  fact  the  pre-Aristotelian  logoi  were 
often  so  understood.  However,  this  interpretation  seems  untenable: 
Aristotle  himself  was  aware  of  the  very  abstract  laws  of  propositional 
logic  only  exceptionally  and  at  the  end  of  his  scientific  career;  so 
much  the  less  ought  we  to  ascribe  this  -  Megarian-Stoic  -  manner  of 
thought  to  the  pre-Aristotelians.  We  have  rather  to  do  with  certain 
specifications  of  general  rules  of  propositional  logic.  Thus  these 
dialecticians  were  not  thinking  of,  for  example,  the  abstract  scheme 
of  propositional  logic  corresponding  to  modus  ponendo  ponens: 

7.101  If  p,  then  q;  but  p ;  therefore  q: 
but  rather  of  the  more  special  law 

7.102  If  A  belongs  to  x,  then  B  also  belongs  to  x;  but  .4. 
belongs  to  x;  therefore  B  also  belongs  to  x. 



We  purposely  omit  quantifiers  here,  since  while  such  were  necessa- 
rily present  to  the  thought  obscurely,  at  this  level  there  can  be  no 
question  of  a  conscious  acceptance  of  such  logical  apparatus. 

We  note  further,  that  at  the  level  of  pre-Aristotelian  dialectic,  it  is 
always  a  matter  of  rules  not  of  laws;  they  are  principles  stating  how 
one  should  proceed,  not  laws,  which  describe  an  objective  state  of 
affairs.  That  does  not  mean  of  course  that  the  dialecticians  were  in 
any  way  conscious  of  the  distinction  between  the  two;  but  from  our 
point  of  view,  what  they  used  were  rules. 

This  said,  we  can  interpret  as  follows  the  several  logoi  previously 
adduced.  For  each  we  give  the  logical  sentence  corresponding  to  the 
rule  of  inference  which  it  employs. 

Zeno  quoted  by  Simplicius  (7.02,  03,  04) : 

7.021  If  A  belongs  to  x  then  B  and  C  also  belong  to  x;  but 
B  and  C  do  not  belong  to  x;  therefore  neither  does  A  belong 
to  x. 

7.022  Suppose  that  if  A  belongs  to  x,  B  also  belongs  to  x 
and  if  B  belongs  to  x,  C  also  belongs  to  x,  then  if  A  belongs  to 
x,  C  also  belongs  to  x. 

Plato  in  the  Theaeletus  (7.05) : 

7.051  If  A  belongs  to  x  then  A  does  not  belong  to  x; 
therefore  A  does  not  belong  to  x. 

Closer  examination  of  that  last  item  shows  that  it  is  much  more 
complex  and  belongs  to  the  realm  of  metalogic.  Plato's  thought 
proceeds  after  this  fashion:  the  proposition  propounded  by  Protago- 
ras means:  for  every  x,  if  x  says  'p',  then  p.  Let  us  abbreviate  that 
by  *S\  Now  there  is  some  (at  least  one)  x  who  says  that  S  is  not  the 
case.  Therefore  S  is  not  the  case.  Therefore  if  S,  then  not  S.  From 
which  it  follows  in  accordance  with  7.051,  that  S  is  not  the  case. 
While  Plato  certainly  did  not  expressly  draw  this  conclusion,  he 
evidently  intended  it. 

Aristotle  quoted  by  Alexander  (7.07)  : 

7.071  Suppose  that  if  A  belongs  to  x,  A  belongs  to  x,  and 
if  A  does  not  belong  to  x,  A  belongs  to  a?,  then  A  belongs  to  x. 

The  anonymous  scholiast  has  a  fuller  formula  (7.09) : 

7.091  If  A  belongs  to  aj,  then  A  belongs  to  x;  if  A  does  not 
belong  to  x,  then  A  belongs  to  x;  either  A  belongs  to  x  or  A 
does  not  belong  to  x;  therefore  A  belongs  to  x, 
but  whether  it  actually  occurred  in  Aristotle  may  be  doubted. 
Possibly  the  Protreplicus  contained  merely  the  simple  formula, 
transmitted  by  Lactantius : 



7.092  If  A  does  not  belong  to  x,  then  A  belongs  to  x; 
therefore  A  belongs  to  x. 

A  series  of  similar  formulae  underlie  the  processes  to  be  found  in 
the  great  Gorgias-fragment  (7.06),  but  these  appear  to  be  so  markedly 
interpreted  in  the  light  of  Stoic  logic  that  we  have  no  guarantee  of 
anything  genuinely  due  to  the  sophist  himself. 

§  8.   PLATO 

While  Plato,  in  respect  of  many  rules  used  in  his  dialectic, 
belongs  to  the  same  period  as  Zeno  (as  too  does  the  youthful  Ari- 
stotle), he  begins  something  essentially  new  in  our  field,  and  that 
from  several  points  of  view. 


In  the  first  place  Plato  rendered  the  immortal  service  of  being  the 
first  to  grasp  and  formulate  a  clear  idea  of  logic.  The  relevant  text 
occurs  in  the  Timaeus  and  runs : 

8.01  God  invented  and  gave  us  sight  to  the  end  that  we 
might  behold  the  courses  of  intelligence  in  the  heaven,  and 
apply  them  to  the  courses  of  our  own  intelligence  which  are 
akin  to  them,  the  unperturbed  to  the  perturbed;  and  that  we, 
learning  them  and  partaking  of  the  natural  truth  of  reason, 
might  imitate  the  absolutely  unerring  courses  of  God  and 
regulate  our  own  vagaries. 

Such  a  conception  of  logic  was,  however,  only  possible  for  Plato, 
because  he  was,  as  it  seems,  the  originator  of  another  quite  original 
idea,  namely  that  of  universally  necessary  laws  (granting  that  he 
depended  in  this  on  the  logos-doctrine  of  Heracleitus  and  other 
earlier  thinkers).  The  concept  of  such  laws  is  closely  connected  with 
Plato's  theory  of  ideas,  which  itself  developed  through  reflection  on 
Geometry  as  it  then  existed.  The  whole  post-Platonic  western 
tradition  is  so  penetrated  with  these  ideas,  that  it  is  not  easy  for  a 
westerner  to  grasp  their  enormous  significance.  Evidently  no  formal 
logic  was  possible  without  the  notion  of  universally  valid  law.  From 
this  point  of  view  the  importance  of  Plato  for  the  history  of  logic 
can  best  be  seen  when  we  consider  the  development  of  the  science  in 
India,  i.e.  in  a  culture  which  had  to  create  logic  without  a  Plato.  One 
can  see  in  the  history  of  Indian  logic  that  it  took  hundreds  of  years 
to  accomplish  what  was  done  in  Greece  in  a  generation  thanks  to 
the  elan  of  Plato's  genius,  namely  to  rise  to  the  standpoint  of  uni- 
versal validity. 



We  cannot  here  expound  Plato's  doctrine  of  ideas,  as  it  belongs  to 
ontology  and  metaphysics,  and  is  further  beset  with  difficult  pro- 
blems of  literary  history. 


Plato  tried  throughout  his  life  to  realize  the  ideal  of  a  logic  as  laid 
down  above,  but  without  success.  The  tollowing  extracts  from  his 
dialectic,  in  which  he  makes  a  laboured  approach  to  quite  simple 
laws,  show  how  difficult  he  found  it  to  solve  logical  questions  that 
seem  elementary  to  us. 

8.02  Socrates:  Then  I  shall  proceed  to  add,  that  if  the 
temperate  soul  is  the  good  soul,  the  soul  which  is  in  the 
opposite  condition,  that  is,  the  foolish  and  intemperate,  is 
the  bad  soul.  -  Very  true.  -  And  will  not  the  temperate  man 
do  what  is  proper,  both  in  relation  to  the  gods  and  to  men;  - 
for  he  would  not  be  temperate  if  he  did  not?  -  Certainly  he 
will  do  what  is  proper. 

8.03  Socrates:  Tell  me,  then,  -  Is  not  that  which  is  pious 
necessarily  just? 

Euthyphro:  Yes. 

Socrates:  And  is,  then,  all  which  is  just  pious?  or,  is  that 
which  is  pious  all  just,  but  that  which  is  just,  only  in  part  and 
not  all,  pious? 

Euthyphro:  I  do  not  understand  you,  Socrates. 

8.04  When  you  asked  me,  I  certainly  did  say  that  the 
courageous  are  the  confident;  but  I  was  never  asked  whether 
the  confident  are  the  courageous;  if  you  had  asked  me,  I 
should  have  answered  'Not  all  of  them :'  and  what  I  did  answer 
you  have  not  proved  to  be  false,  although  you  proceeded  to 
show  that  those  who  have  knowledge  are  more  courageous 
than  they  were  before  they  had  knowledge,  and  more  coura- 
geous than  others  who  have  no  knowledge,  and  were  then  led 
on  to  think  that  courage  is  the  same  as  wisdom.  But  in  this 
way  of  arguing  you  might  come  to  imagine  that  strength  is 
wisdom.  You  might  begin  by  asking  whether  the  strong  are 
able,  and  I  should  say  'Yes';  and  then  whether  those  who 
know  how  to  wrestle  are  not  more  able  to  wrestle  than  those 
who  do  not  know  how  to  wrestle,  and  more  able  after  than 
before  they  had  learned,  and  I  should  assent.  And  when  I  had 
admitted  this,  you  might  use  my  admissions  in  such  a  way  as 
to  prove  that  upon  my  view  wisdom  is  strength;  whereas 
in  that  case  I  should  not  have  admitted,  any  more  than  in  the 



other,  that  the  able  are  strong,  although  I  have  admitted  that 

the  strong  are  able.  For  there  is  a  difference  between  ability 
and  strength;  the  former  is  given  by  knowledge  as  well  as  by 
madness  or  rage,  but  strength  comes  from  nature  and  a 
healthy  state  of  the  body.  And  in  like  manner  J  say  of  confi- 
dence and  courage,  that  they  are  not  the  same;  and  I  argue 
that  the  courageous  are  confident,  but  not  all  the  confident 
courageous.  For  confidence  may  be  given  to  men  by  art,  and 
also,  like  ability,  by  madness  and  rage;  but  courage  comes  to 
them  from  nature  and  the  healthy  state  of  the  soul. 

In  the  first  of  these  texts  is  involved  the  (false)  thesis:  Suppose,  if 
A  belongs  to  x,  B  also  belongs  to  x,  then:  if  A  does  not  belong  to  x, 
then  B  does  not  belong  to  x.  The  second  shows  the  difficulties  found 
concerning  the  convertibility  of  universal  affirmative  sentences:  viz. 
whether  'all  B  is  A'  follows  from  'all  A  is  B ' .  The  third  text  shows 
still  more  clearly  how  hard  Plato  felt  these  questions  to  be;  it  further 
has  the  great  interest  that,  to  show  the  invalidity  of  the  fore- 
going rule  of  conversion,  he  betakes  himself  to  complicated  extra- 
logical  discussions  -  about  bodily  strength,  for  instance. 


Yet  Plato's  approximations  were  not  without  fruit.  He  seems  to 
have  been  the  first  to  progress  from  a  negative  dialectic  to  the  con- 
cept of  positive  proof;  for  him  the  aim  of  dialectic  is  not  to  refute 
the  opinions  of  opponents  but  positive  'definition  of  the  essence'.  In 
this  he  definitely  directed  attention  to  the  logic  of  predicates,  which 
is  probably  the  cause  of  Aristotelian  logic  taking  the  form  it  did. 
The  chief  goal  which  Plato  set  himself  was  to  discover  essences,  i.e.  to 
find  statements  which  between  them  define  what  an  object  is.  For 
this  he  found  a  special  method  -  the  first  logical,  consciously  ela- 
borated inferential  procedure  known  to  us  -  namely  his  famous 
'hunt'  for  the  definition  by  division  (Sioctpeais).  How  thoroughly 
conscious  he  was  of  not  only  using  such  a  method  but  of  endeavour- 
ing to  give  it  the  clearest  possible  formulation,  we  see  in  the  cele- 
brated text  of  the  Sophist  in  which  the  method,  before  being  practis- 
ed, is  applied  in  an  easy  example: 

8.05  Stranger:  Meanwhile  you  and  I  will  begin  together  and 
enquire  into  the  nature  of  the  Sophist,  first  of  the  three :  I 
should  like  you  to  make  out  what  he  is  and  bring  him  to  light 
in  a  discussion;  for  at  present  we  are  only  agreed  about  the 
name,  but  of  the  thing  to  which  we  both  apply  the  name 
possibly  you  have  one  notion  and  I  another;  whereas  we  ought 
always  to  come  to  an  understanding  about  the  thing  itself  in 



terms  of  a  definition,  and  not  merely  about  the  name  minus  the 
definition.  Now  the  tribe  of  Sophists  which  we  are  investi- 
gation is  not  easily  caught  or  defined ;  and  the  world  has  long 
ago  agreed,  that  if  great  subjects  are  to  be  adequately  treated, 
they  must  be  studied  in  the  lesser  and  easier  instances  of  them 
before  we  proceed  to  the  greatest  of  all.  And  as  I  know  that  the 
tribe  of  Sophists  is  troublesome  and  hard  to  be  caught,  I 
should  recommend  that  we  practise  beforehand  the  method 
which  is  to  be  applied  to  him  on  some  simple  and  smaller  thing, 
unless  you  can  suggest  a  better  way. 

Theaetetus:  Indeed  I  cannot. 

Stranger:  Then  suppose  that  we  work  out  some  lesser 
example  which  will  be  a  pattern  of  the  greater? 

Theaetetus :  Good. 

Stranger:  What  is  there  which  is  well  known  and  not  great, 
and  is  yet  as  susceptible  of  definition  as  any  larger  thing?  Shall 
I  say  an  angler?  He  is  familiar  to  all  of  us,  and  not  a  very 
interesting  or  important  person. 

Theaetetus :  He  is  not. 

Stranger:  Yet  I  suspect  that  he  will  furnish  us  with  the  sort 
of  definition  and  line  of  enquiry  which  we  want. 

Theaetetus:  Very  good. 

Stranger:  Let  us  begin  by  asking  whether  he  is  a  man  having 
art  or  not  having  art,  but  some  other  power. 

Theaetetus:  He  is  clearly  a  man  of  art. 

Stranger:  And  of  arts  there  are  two  kinds? 

Stranger:  Seeing,  then,  that  all  arts  are  either  acquisitive  or 
creative,  in  which  class  shall  we  place  the  art  of  the  angler? 

Theaetetus :  Clearly  in  the  acquisitive  class. 

Stranger:  And  the  acquisitive  may  be  subdivided  into  two 
parts:  there  is  exchange,  which  is  voluntary  and  is  effected  by 
gifts,  hire,  purchase;  and  the  other  part  of  acquisitive,  which 
takes  by  force  of  word  or  deed,  may  be  termed  conquest? 

Theaetetus:  That  is  implied  in  what  has  been  said. 

Stranger:  And  may  not  conquest  be  again  subdivided? 

Theaetetus :  How? 

Stranger:  Open  force  may  be  called  fighting,  and  secret 
force  may  have  the  general  name  of  hunting? 

Theaetetus :  Yes. 

Stranger:  And  there  is  no  reason  why  the  art  of  hunting 
should  not  be  further  divided. 



Theaetetus :  How  would  you  make  the  division? 

Stranger:  Into  the  hunting  of  living  and  of  lifeless  prey. 

Theaetetus  :Yes,  if  both  kinds  exist. 

Stranger:  Of  course  they  exist;  but  the  hunting  after  life- 
less things  having  no  special  name,  except  some  sorts  of 
diving,  and  other  small  matters,  may  be  omitted;  the  hunting 
after  living  things  may  be  called  animal  hunting. 

Theaetetus :  Yes. 

Stranger:  And  animal  hunting  may  be  truly  said  to  have  two 
divisions,  land-animal  hunting,  which  has  many  kinds  and 
names,  and  water-animal  hunting,  or  the  hunting  after  ani- 
mals who  swim? 

Theaetetus:  True. 

Stranger:  And  of  swimming  animals,  one  class  lives  on  the 
wing  and  the  other  in  the  water? 

Theaetetus :  Certainly. 

Stranger:  Fowling  is  the  general  term  under  which  the 
hunting  of  all  birds  is  included. 

Theaetetus :  True. 

Stranger:  The  hunting  of  animals  who  live  in  the  water  has 
the  general  name  of  fishing. 

Theaetetus :  Yes. 

Stranger:  And  this  sort  of  hunting  may  be  further  divided 
also  into  two  principal  kinds? 

Theaetetus :  What  are  they? 

Stranger:  There  is  one  kind  which  takes  them  in  nets,  another 
which  takes  them  by  a  blow. 

Theaetetus:  What  do  you  mean,  and  how  do  you  distinguish 
them  ? 

Stranger :  As  to  the  first  kind  -  all  that  surrounds  and  encloses 
anything  to  prevent  egress,  may  be  rightly  called  an  enclosure. 

Theaetetus:  Very  true. 

Stranger:  For  which  reason  twig  baskets,  casting-nets, 
nooses,  creels,  and  the  like  may  all  be  termed  'enclosures'? 

Theaetetus:  True. 

Stranger:  And  therefore  this  first  kind  of  capture  may  be 
called  by  us  capture  with  enclosures,  or  something  of  that  sort  ? 

Theaetetus:  Yes. 

Stranger:  The  other  kind,  which  is  practised  by  a  blow  with 
hooks  and  three-pronged  spears,  when  summed  up  under  one 
name,  may  be  called  striking,  unless  you,  Theaetetus,  can 
find  some  better  name  ? 



Theaetetus :  Never  mind  the  name  -  what  you  suggest  will 
do  very  well. 

Stranger:  There  is  one  mode  of  striking,  which  is  done  at 
night,  and  by  the  light  of  a  fire,  and  is  by  the  hunters  them- 
selves called  firing,  or  spearing  by  firelight. 

Theaetetus:  True. 

Stranger:  And  the  fishing  by  day  is  called  by  the  general 
name  of  barbing,  because  the  spears,  too,  are  barbed  at  the 

Theaetetus:  Yes,  that  is  the  term. 

Stranger:  Of  this  barb-fishing,  that  which  strikes  the  fish 
who  is  below  from  above  is  called  spearing,  because  this  is 
the  way  in  which  the  three-pronged  spears  are  mostly  used. 

Theaetetus :  Yes,  it  is  often  called  so. 

Stranger:  Then  now  there  is  only  one  kind  remaining. 

Theaetetus :  What  is  that  ? 

Stranger:  When  a  hook  is  used,  and  the  fish  is  not  struck  in 
any  chance  part  of  his  body,  as  he  is  with  the  spear,  but  only 
about  the  head  and  mouth,  and  is  then  drawn  out  from  below 
upwards  with  reeds  and  rods:  -  What  is  the  right  name  of 
that  mode  of  fishing,  Theaetetus? 

Theaetetus :  I  suspect  that  we  have  now  discovered  the  object 
of  our  search. 

Stranger:  Then  now  you  and  I  have  come  to  an  under- 
standing not  only  about  the  name  of  the  angler's  art,  but 
about  the  definition  of  the  thing  itself.  One  half  of  all  art  was 
acquisitive  -  half  of  the  acquisitive  art  was  conquest  or 
taking  by  force,  half  of  this  was  hunting,  and  half  of  hunting 
was  hunting  animals,  half  of  this  was  hunting  water  animals  - 
of  this  again,  the  under  half  was  fishing,  half  of  fishing  was 
striking;  a  part  of  striking  was  fishing  with  a  barb,  and  one 
half  of  this  again,  being  the  kind  which  strikes  with  a  hook 
and  draws  the  fish  from  below  upwards,  is  the  art  which  we 
have  been  seeking,  and  which  from  the  nature  of  the  opera- 
tion is  denoted  angling  or  drawing  up. 

Theaetetus:  The  result  has  been  quite  satisfactorily  brought 

The  process  is  evidently  not  conclusive:  as  Aristotle  has  forcibly 
shown  (8.06),  it  involves  a  succession  of  assertions,  not  a  proof;  it 
may  be  helpful  as  a  method,  but  it  is  not  formal  logic. 

Formal  logic  is  reserved  for  Aristotle.  But  a  close  examination  of 
the  contents  of  his  logical  works  assures  us  that  everything  contained 



in  the  Organon  is  conditioned  in  one  way  or  another  by  the  practice 
of  Platonism.  The  Topics  is  probably  only  a  conscious  elaboration  of 
the  numerous  logoi  current  in  the  Academy;  even  the  Analytics, 
invention  of  Aristotle's  own  as  it  was,  is  evidently  based  on 
'division',  which  it  improved  and  raised  to  the  level  of  a  genuine 
logical  process.  That  is  the  second  great  service  which  Plato 
rendered  to  formal  logic:  his  thought  made  possible  the  emergence 
of  the  science  with  Aristotle. 




The  surviving  logical  works  of  Aristotle  set  many  difficult  pro- 
blems of  literary  history  which  as  yet  are  only  partly  solved.  They  are 
of  outstanding  importance  for  the  history  of  the  problems  of  logic, 
since  within  the  short  span  of  Aristotle's  life  formal  logic  seems  to 
have  made  more  progress  than  in  any  other  epoch.  It  is  no  exagge- 
ration to  say  that  Aristotle  has  a  unique  place  in  the  history  of 
logic  in  that  1.  he  was  the  first  formal  logician,  2.  he  developed  formal 
logic  in  at  least  two  (perhaps  three)  different  forms,  3.  he  consciously 
elaborated  some  parts  of  it  in  a  remarkably  complete  way.  Further- 
more, he  exercised  a  decisive  influence  on  the  history  of  logic  for 
more  than  two  thousand  years,  and  even  today  much  of  the  doctrine 
is  traceable  back  to  him.  It  follows  that  an  adequate  understanding 
of  the  development  of  his  logical  thought  is  of  extreme  importance 
for  an  appreciation  of  the  history  of  logical  problems  in  general,  and 
particularly  of  course  for  western  logic. 


The  surviving  works  of  the  Stagirite  were  set  in  order  and  edited 
by  Andronicus  of  Rhodes  in  the  first  century  b.c.  The  resulting 
Corpus  Aristotelicum  contains,  as  to  logical  works,  first  and  fore- 
most what  was  later  called  the  Organon,  comprising: 

1.  The  Categories, 

2.  About  Propositions  (properly:  About  Interpretation;  we  shall 
use  the  title  Hermeneia), 

3.  The  Prior  Analytics,  two  books:  A  and  B, 

4.  The  Posterior  Analytics,  two  books :  A  and  B, 

5.  The  Topics,  eight  books:  A,B,  T,  A,  E,  Z,  H,  0, 

6.  The  Sophistic  Refutations,  one  book. 

Besides  these,  the  whole  fourth  book  (T)  of  the  so-called  Meta- 
physics is  concerned  with  logical  problems,  while  other  works,  e.g.  the 
Rhetoric  and  Poetics  contain  occasional  points  of  logic. 


The  most  important  problems  concerning  the  Organon  are  the 
following : 

1.  Authenticity 

In  the  past  the  genuineness  of  all  Aristotle's  logical  writings  has 
often  been  doubted.  Today,  apart  from  isolated  passages  and 
perhaps  individual  chapters,  the  Categories  alone  is  seriously  con- 
sidered to  be  spurious.  The  doubt  about  the  genuineness  of  the 


A  R  I  S  T  O  T  L  E 

Hermeneia  seems  not  convincing.  The  remaining  works  rank  by  and 
large  as  genuine.  * 

2.  Character 

Should  we  view  the  logical  works  of  Aristotle  as  methodically 
constructed  and  systematic  treatises?  Researches  made  hitherto 
allow  us  to  suppose  this  only  for  some  parts  of  the  Organon.  The 
Hermeneia  and  Topics  enjoy  the  relatively  greatest  unity.  The 
Prior  Analytics  are  evidently  composed  of  several  strata,  while  the 
Posterior  Analytics  are  mainly  rather  a  collection  of  notes  for  lectures 
than  a  systematic  work.  But  even  in  those  parts  of  the  Organon  that 
are  systematically  constructed  later  additions  are  to  be  found  here 
and  there. 

3.  Chronology 

The  Organon,  arranged  as  we  have  it,  is  constructed  on  a  syste- 
matic principle:  the  Categories  treats  of  terms,  the  Hermeneia  of 
propositions,  the  remaining  works  of  inference:  thus  the  Prior 
Analytics  treats  of  syllogisms  in  general,  the  three  other  works 
successively  of  apodeictic  (scientific),  dialectical  and  sophistical 
syllogisms.  For  this  systematization  Andronicus  found  support  in 
the  very  text  of  the  Organon;  e.g.  at  the  beginning  of  the  Prior 
Analytics  it  is  said  (9.02)  that  the  syllogism  consists  of  propositions 
(npOTCcaic,) ,  these  of  terms  (6po«;).  In  the  Topics  (9.03)  and  also  in 
the  Prior  Analytics  (9.04)  the  syllogism  is  analysed  in  just  that  way. 
At  the  end  of  the  Sophistic  Refutations  occurs  the  sentence  already 
cited  (7.01),  which  appears  to  indicate  that  this  work  is  the  latest 
of  Aristotle's  logical  works. 

It  is  also  not  impossible  that  at  the  end  of  his  life  Aristotle  himself 
drafted  an  arrangement  of  his  logic  and  accordingly  ordered  his 
notes  and  treatises  somewhat  as  follows.  But  of  course  this  late 
systematization  has,  to  our  present  knowledge,  little  to  do  with  the 
actual  development  of  this  logic. 

We  have  no  extrinsic  criteria  to  help  us  establish  the  chronolo- 
gical sequence  of  the  different  parts  of  the  Organon.  On  the  other 
hand  their  content  affords  some  assistance,  as  will  now  be  briefly 

*  The  thesis  of  Josef  Ziircher  (9.01)  that  nearly  all  the  formal  logic  in  the 
Organon  is  due  not  to  Aristotle  but  to  his  pupil  Theophrastus,  is  not  worth 
serious  consideration. 

*  *  Chr.  Brandis  opened  up  the  great  matter  of  the  literary  problems  of  the 
Organon  in  his  paper  Vber  die  Reihenfolge  des  aristotelischen  Organons  (9.05);  the 
well-known  work  of  W.  Jager  (9.06)  contributed  important  insights;  its  basic 
pre-suppositions  were  applied  to  the  Organon  by  F.  Solmsen  (9.07).  Solmsen's 
opinions  were  submitted  to  a  thorough  criticism  by  Sir  W.  D.  Ross  (9.08)  with 
an  adverse  result  in  some  cases.  Important  contributions  to  the  chronology  of 
the  Organon  are  to  be  found  in  A.  Becker  (9.09)  and  J.  Lukasiewicz  ^9.10). 



a.  Chronological  criteria 

aa)  A  first  criterion  to  determine  the  relative  date  of  origin  is 
afforded  by  the  fact  that  the  syllogism  in  the  sense  of  the  Prior 
Analytics  (we  shall  call  it  the  'analytical  syllogism')  is  completely 
absent  from  several  parts  of  the  Organon.  But  it  is  one  of  the  most 
important  discoveries,  and  it  can  hardly  be  imagined  that  Aristotle 
would  have  failed  to  make  use  of  it,  once  he  had  made  it.  We 
conclude  that  works  in  which  there  are  no  analytic  syllogisms  are 
earlier  than  those  in  which  they  occur. 

bb)  In  some  parts  of  the  Organon  we  find  variables  (viz.  the  letters 
A,  B,  r,  etc.),  in  others  not.  But  variables  are  another  epoch- 
making  discovery  in  the  domain  of  logic,  and  the  degree  to  which  it 
impressed  Aristotle  can  be  seen  in  the  places  where  he  uses  and 
abuses  them  to  the  point  of  tediousness.  Now  there  are  some  works 
where  variables  would  be  very  useful  but  where  they  do  not  occur. 
We  suppose  that  these  works  are  earlier  than  those  where  they  do 

cc)  The  third  criterion  -  afforded  by  the  technical  level  of  the 
thought  -  cannot,  unlike  the  first  two,  be  formulated  simply,  but  is 
apparent  to  every  experienced  logician  at  his  first  perusal  of  a  text. 
From  this  point  of  view  there  are  big  differences  between  the 
various  passages  of  the  Organon:  in  some  we  find  ourselves  at  a  still 
very  primitive  level,  reminiscent  of  pre-Socratic  logic,  while  in 
others  Aristotle  shows  himself  to  be  the  master  of  a  strictly  formal 
and  very  pure  logical  technique.  One  aspect  of  this  progress  appears 
in  the  constantly  developing  analysis  of  statements:  at  first  this 
is  accomplished  by  means  of  the  simple  subject-predicate  schema 
(S  —  P),  then  quantifiers  occur  (lP  belongs  [does  not  belong]  to  all, 
to  none,  to  some  S'),  finally  we  meet  a  subtle  formula  that  reminds 
us  of  the  modern  formal  implication :  'All  that  belongs  to  S,  belongs 
also  to  P\  This  criterion  can  be  formulated  as  follows:  the  higher 
and  more  formal  the  technique  of  analysis  and  proof,  so  much  the 
later  is  the  work. 

dd)  Modal  logic  corresponds  much  better  with  Aristotle's  own 
philosophy  (which  contains  the  doctrine  of  act  and  potency  as  an 
essential  feature)  than  does  purely  assertoric  logic  in  which  the 
distinction  between  act  and  potency  obtains  no  expression.  Asser- 
toric logic  fits  much  better  with  the  Platonism  to  which  Aristotle 
subscribed  in  his  youth.  Accordingly  we  may  view  those  writings 
and  chapters  containing  modal  logic  as  having  been  composed  later. 

ee)  Some  of  these  criteria  can  be  further  sharpened.  Thus  we  can 
trace  some  development  in  the  theory  of  analytic  syllogism.  Again, 
Aristotle  seems  to  have  used  letters  at  first  as  mere  abbreviations 
for  words  and  only  later  as  genuine  variables.  Finally  one  can  detect 
a  not  insignificant  progress  in  the  structure  of  modal  logic. 

It  may  certainly  be  doubted  whether  any  one  of  these  criteria  is 



of  value  by  itself  for  establishing  the  chronology.  But  when  all,  or 
at  least  several  of  them  point  in  the  same  direction,  the  resulting 
sequence  seems  to  enjoy  as  high  a  degree  of  probability  as  is  ever 
possible  in  the  historical  sciences. 

b.  Chronological  list 

The  application  of  these  criteria  enable  us  to  draw  up  the  following 

chronological  list  of  Aristotle's  logical  writings: 

aa)  The  Topics  (together  with  the  Categories  if  this  is  to  be  accep- 
ted as  genuine)  undoubtedly  comes  at  the  start.  There  is  to  be  found 
in  it  no  trace  of  the  analytic  syllogism,  no  variables,  no  modal  logic, 
and  the  technical  level  of  the  thought  is  relatively  low.  While  the 
Sophistic  Refutations  simply  forms  the  last  book  of  the  Topics,  it 
appears  to  have  been  composed  a  little  later.  Book  F  of  the  Meta- 
physics probably  belongs  to  the  same  period.  The  Topics  and  the 
Refutations  together  contain  Aristotle's  first  logic.  The  remark  at 
the  end  of  the  Sophistic  Refutations  about  the  'whole'  of  logic 
refers  to  that  elaboration. 

bb)  The  Hermeneia  and  -  perhaps  -  book  B  of  the  Posterior 
Analytics  form  a  kind  of  transitional  stage:  the  syllogistic  can  be 
seen  emerging.  In  the  Hermeneia  we  hear  nothing  of  syllogism  and 
there  are  no  variables.  Both,  but  evidently  only  in  an  early  stage 
of  development,  occur  in  Posterior  Analytics  book  B.  The  technical 
level  of  thought  is  much  higher  than  in  the  Topics.  The  Hermeneia 
also  contains  a  doctrine  of  modality,  which  is,  however,  quite 
primitive  compared  with  that  in  the  Prior  Analytics. 

cc)  Book  A  of  the  Prior  Analytics,  with  the  exception  of  chapters 
8-22,  contains  Aristotle's  second  logic,  a  fully  developed  assertoric 
syllogistic.  He  is  by  now  in  possession  of  a  clear  idea  of  analytic 
syllogism,  uses  variables  with  sureness,  and  moves  freely  at  a  rela- 
tively high  technical  level.  The  analysis  of  propositions  has  been 
deepened.  Missing,  as  yet,  are  modal  logic,  and  reflective  considera- 
tion of  the  syllogistic  system.  Perhaps  book  A  of  the  Posterior 
Analytics  may  be  ascribed  to  the  same  period.  Solmsen  made  this 
the  first  of  all  the  analytic  books,  but  W.  D.  Ross's  arguments 
against  this  seem  convincing.  (The  latter  holds  that  book  B  of  the 
Posterior  Analytics  is  also  later  than  the  Prior.) 

dd)  Finally  we  may  ascribe  to  a  still  later  period  chapters  8-22  of 
book  A,  which  contains  the  modal  syllogistic  logic,  and  book  B  of 
the  Prior  Analytics.  These  can  be  said  to  contain  Aristotle's  third 
logic,  which  differs  less  from  the  second  than  does  the  second  from 
the  first.  We  find  here  a  developed  modal  logic,  marred  admittedly 
by  many  incompletenesses  and  evidently  not  finished,  and  also 
penetrating    remarks,    partly    metalogical,    about    the    system    of 



syllogistic.  In  them  Aristotle  offers  us  insights  into  formal  logic  of 
remarkable  subtlety  and  acuteness.  He  states  too  some  theorems 
of  propositional  logic  with  the  aid  of  propositional  variables. 

Of  course  there  can  be  no  question  of  absolute  certainty  in  answer- 
ing the  chronological  questions,  especially  as  the  text  is  corrupt  in 
many  places  or  sprinkled  with  bits  from  other  periods.  It  is  only 
certain  that  the  Topics  and  Sophistic  Refutations  contain  a  different 
and  earlier  logic  than  the  Analytics,  and  that  the  Hermeneia  exhi- 
bits an  intermediate  stage.  For  the  rest  we  have  well-founded 
hypotheses  which  can  lay  claim  at  least  to  great  probability. 

In  accordance  with  these  hypotheses  we  shall  speak  of  three 
logics  of  Aristotle. 



Aristotle  has  no  special  technical  name  for  logic:  what  we  now 
call  'logical'  he  calls  'analytic'  (ocvocXutlxo^:  10.01)  or  'following 
from  the  premisses'  (ex  tgW  xeiuivcov:  10.02),  while  the  expression 
'logical'  (Xoyix6<;)  means  the  same  as  our  'probable'  (10.03)  or 
again  'epistemological'. 

10.04  Of  propositions  and  problems  there  are  -  to  com- 
prehend the  matter  in  outline  -  three  divisions :  for  some  are 
ethical  propositions,  some  are  on  natural  philosophy,  while 
some  are  logical.  .  .  such  as  this  are  logical,  e.g.  'Is  the  know- 
ledge of  opposites  the  same  or  not?' 

The  question  whether  logic  is  a  part  of  philosophy  or  its  instrument 
(opyavov)  -  and  hence  an  art  -  is  nowhere  raised  by  Aristotle  in  the 
extant  works. 


Yet  Aristotle  knew  well  enough  what  he  demanded  of  logic. 
That  appears  from  the  model  statements  of  the  subject-matter  of 
his  logical  treatises.  For  instance  he  says  in  the  Topics: 

10.05  First  then  we  must  say  what  reasoning  is,  and  what 
its  varieties  are,  in  order  to  grasp  dialectical  reasoning:  for 
this  is  the  object  of  our  search  in  the  treatise  before  us.  Now 
reasoning  is  an  argument  in  which,  certain  things  being  laid 
down,  something  other  than  these  necessarily  comes  about 
through  them.  It  is  a  'demonstration'  when  the  premisses 
from  which  the  reasoning  starts  are  true  and  primary  .  .   . 



reasoning,  on  the  other  hand,  is  'dialectical',  if  it  reasons 
from  opinions  that  are  generally  accepted.  .  .  .  Again,  reason- 
ing is  'contentious'  if  it  starts  from  opinions  that  seem  to  be 
generally  accepted,  but  are  not  really  such,  or  again  if  it 
merely  seems  to  reason  from  opinions  that  are  or  seem  to  be 
generally  accepted. 

Compare  the  following  text  from  the  Prior  Analytics: 

10.06  After  these  distinctions  we  now  state  by  what  means, 
when,  and  how  every  syllogism  is  produced;  subsequently 
we  must  speak  of  demonstration.  Syllogism  should  be  discuss- 
ed before  demonstration,  because  syllogism  is  the  more 
general:  demonstration  is  a  sort  of  syllogism,  but  not  every 
syllogism  is  a  demonstration. 

The  thought  is  perfectly  clear:  Aristotle  is  looking  for  relations 
of  dependence  which  authorize  necessary  inference,  and  in  that 
connection  makes  a  sharp  distinction  between  the  validity  of  this 
relation  and  the  kind  of  premisses,  or  their  truth.  The  text  contains 
what  is  historically  the  first  formulation  of  the  concept  of  a  formal 
logic,  universally  valid  and  independent  of  subject-matter. 

Accordingly  it  is  syllogism  which  is  the  subject  of  logic.  This  is 
a  form  of  speech  (Xoyo^)  consisting  of  premisses  (nporuikaeu;) 
themselves  composed  of  terms  (5poi).  'Premiss'  and  'term'  are  thus 
defined  by  Aristotle: 

10.07  A  premiss  is  a  form  of  speech  which  affirms  or 
denies  something  of  something.  ...  A  term  I  call  that  into 
which  the  premiss  is  resolved,  that  is  to  say  what  is  predicated 
and  that  of  which  it  is  predicated  by  means  of  the  addition 
of  being  or  not  being. 

What  emerges  from  that  text  is  the  complete  neutrality  of  the 
technical  expressions  'term',  'premiss',  'syllogism',  relative  to  any 
philosophical  interpretation.  For  the  premiss  consists  of  terms,  the 
syllogism  of  premisses,  and  premisses  are  logoi,  which  can  equally 
well  mean  utterances  or  thoughts  or  objective  contents,  so  that  the 
way  is  open  to  a  formalist,  psychological  or  objectivist  interpreta- 
tion. All  these  interpretations  are  permissible  in  regard  to  Aristo- 
telian logic;  the  purely  logical  system  excludes  none  of  them.  Guided 
by  his  original  intuition  the  founder  of  formal  logic  so  chose  his 
terminology  as  to  rise  above  the  clash  of  interpretations  to  the  level 
of  pure  logic. 

However  if  one  considers  Aristotelian  logic  in  its  entirety,  it  is 
easy  to  see  that  this  neutrality  is  not  the  result  of  a  lack  of  interest  in 
problems  of  interpretation,  but  is  on  the  contrary  an  abstraction 



from  a  complex  semiotic  doctrine.  In  some  places  Aristotle  seems  to 
plead  for  a  psychological  type  of  theory,  as  when,  for  example,  he 


10.08  All  syllogism  and  therefore  a  fortiori  demonstration, 
is  addressed  not  to  outward  speech  but  to  that  within  the 

At  the  same  time  it  must  be  said  that  he  attaches  great  importance 
to  the  'outward  speech',  since  he  elaborates  a  well-developed 
theory  of  logical  syntax  and  many  points  of  semantic  interest. 
All  this  teaching,  which  is  next  to  be  considered,  warrants  the  con- 
clusion that  the  practice  of  Aristotelian  logic  was  undoubtedly  to 
regard  meaningful  words  as  its  subject-matter. 


Aristotle  is  the  founder  of  logical  syntax,  following  here  some 
hints  of  the  Sophists  and  Plato.  He  sketched  the  first  attempt  known 
to  us  at  a  system  of  syntactical  categories.  For  we  find  in  the  Her- 
meneia  an  explicit  division  of  the  parts  of  speech  into  atomic  (nouns 
and  verbs)  and  molecular  (sentences). 

10.09  By  a  noun  we  mean  a  sound  significant  by  con- 
vention, which  has  no  reference  to  time,  and  of  which  no 
part  is  significant  apart  from  the  rest. 

10.10  A  verb  is  that  which,  in  addition  to  its  proper 
meaning,  carries  with  it  the  notion  of  time.  No  part  of  it  has 
any  independent  meaning,  and  it  is  a  sign  of  something  said 
of  something  else. 

This  theory  is  further  supplemented  by  discussions  of  cases  and 
inflexions  of  words,  and  by  considerations  about  negated  nouns  and 

10.11  A  sentence  is  a  significant  portion  of  speech,  some 
parts  of  which  have  an  independent  meaning,  that  is  to  say, 
as  an  utterance  (^olgiq) ,  though  not  as  the  expression  of  any 
positive  judgement  (xaTowpaais). 

10.12  The  first  unified  declarative  sentence  is  the  affirma- 
tion; the  next,  the  denial.  All  other  sentences  are  unified 
by  combination. 

Anticipating  the  further  explanations,  we  may  summarize  the 
whole  scheme  of  syntactical  categories  as  presented  in  the  Herme- 
neia,  after  this  fashion : 


A  FU  S  T  O  T  L  E 







(6vo[i.a)  in 

other  (poLGZic, 


noun  in 


sense  (10.10) 





negated  noun  (10.19) 
cases  of  noun  (10.20) 
verb  p7)[i.a  (10.21) 
negated  verb  (10.22) 
inflexions  of  verbs  (10.23) 







with  singu- 
lar subject 




taken  not  taken 

universally  universallv 

molecular  (10.26) 

other  Xoyoi,  (also  called  cpfxaeic,)  (10.26) 

This  schema  underlies  the  whole  development  of  logical  syntax, 
and  semantics  too,  until  the  rise  of  mathematical  logic.  Only  this 
last  will  introduce  anything  essentially  new:  the  attempt  to  treat 
syntactical  categories  by  means  of  an  artificial  language.  Aristotle 
on  the  other  hand,  and  with  him  the  Stoic  and  scholastic  traditions, 
sought  to  grasp  the  syntactical  structure  of  ordinary  language. 


The  texts  previously  cited  already  contain  points  belonging  to 
the  domain  of  semantics.  The  general  principle  is  thus  formulated  by 



10.28  Spoken  words  are  the  symbols  of  mental  experience 
and  written  words  are  the  symbols  of  spoken  words. 

It  follows  that  thoughts  are  themselves  symbols  of  things. 
Aristotle  lays  great  stress  on  the  parallelism  between  things, 
thoughts  and  symbols,  and  correspondingly  develops  two  important 
semiotic  theories: 

10.29  Things  are  said  to  be  named  'equivocally'  when, 
though  they  have  a  common  name,  the  definitions  correspond- 
ing with  the  name  differs  for  each.  Thus  a  real  man  and  a 
figure  in  a  picture  can  both  lay  claim  to  the  name  'animal'.  .  .  . 

Things  are  said  to  be  named  'univocally'  which  have  both 
the  name  and  the  definition  answering  to  the  name  in  com- 
mon. A  man  and  an  ox  are  both  'animal'  .... 

Things  are  said  to  be  named  'derivatively',  which  derive 
their  name  from  some  other  name,  but  differ  from  it  in 
termination.  Thus  the  grammarian  derives  his  name  from  the 
word  'grammar'  and  the  courageous  man  from  the  word 

Equivocity  must  be  excluded  from  demonstrations,  since  it 
leads  to  fallacies  (10.30).  Elsewhere  Aristotle  distinguishes  various 
kinds  of  equivocity. 

10.31  The  good,  therefore,  is  not  some  common  element 
answering  to  one  Idea.  But  what  then  do  we  mean  by  the 
good  ?  It  is  surely  not  like  the  things  that  only  chance  to  have 
the  same  name.  Are  goods  one,  then,  by  being  derived  from 
one  good  or  by  all  contributing  to  one  good,  or  are  they 
rather  one  by  analogy?  Certainly  as  sight  is  in  the  body,  so 
is  reason  in  the  soul,  and  so  on  in  other  cases. 

This  division  can  be  presented  schematically  as  follows: 


in  strict  sense 
dcTco  tux"/)*; 
(accidentally  equivocal 

in  broad  sense 
(systematically  equivocal) 

from  one  (deep*  hoc,) 
to  one  (npbq  lv) 

by  proportion 

(xoct'  avaXoytav) 



In  the  Metaphysics  and  Hermeneia  we  find  a  clear  sern ioric 
theory  of  truth : 

10.32  For  falsity  and  truth  are  not  in  things  -  it  is  not  as  if 
the  good  were  true  and  the  bad  were  in  itself  false  -  but  in 

10.33  As  there  are  in  the  mind  thoughts  which  do  not 
involve  truth  or  falsity,  and  also  those  which  must  be  either 
true  or  false,  so  it  is  in  speech.  For  truth  and  falsity  imply  com- 
bination and  separation.  Nouns  and  verbs,  provided  nothing 
is  added,  are  like  thoughts  without  combination  or  separation. 

10.34  Yet  not  every  sentence  states  something,  but  only 
those  in  which  there  is  truth  or  falsity,  and  not  all  are  of 
that  kind.  Thus  a  prayer  is  a  sentence,  but  is  neither  true 
nor  false  .  .  .  the  present  theory  is  concerned  with  such  sen- 
tences as  are  statements  (#7co<pocvTix&s  X6yo<;). 

Aristotle  also  constructs  a  definition  of  truth  by  means  of  equi- 

10.35  If  it  is  true  to  say  that  a  thing  is  white,  it  must 
necessarily  be  white;  if  the  reverse  proposition  is  true,  it 
will  of  necessity  not  be  white.  Again,  if  it  is  white,  the  proposi- 
tion stating  that  it  is  white  was  true;  if  it  is  not  white,  the 
proposition  to  the  opposite  effect  was  true.  And  if  it  is  not 
white ,  the  man  who  states  that  it  is  is  making  a  false  statement ; 
and  if  the  man  who  states  that  it  is  white  is  making  a  false 
statement,  it  follows  that  it  is  not  white. 



The  Topics  contains  Aristotle's  first  logic,  and  so  the  first  attempt 
at  a  systematic  presentation  of  our  science.  We  cannot  here  do 
more  than  glance  over  the  mass  of  rules  contained  in  this  work, 
giving  only  its  purpose,  a  discussion  of  the  analysis  of  statements 
as  made  by  Aristotle  in  this  early  work,  and  a  brief  review  of  his 
teaching  about  fallacies.  The  most  important  of  the  formal  rules 
and  laws  of  inference  occurring  here,  continued,  as  it  seems,  to  be 
recognized  as  valid  in  the  later  works,  and  they  will  therefore  be 
considered  in  the  section  on  the  non-analytical  formulae  (§  16). 



11.01  Next  in  order  after  the  foregoing,  we  must  say  for 
how  many  and  for  what  purposes  the  treatise  is  useful.  They 
are  three  -  intellectual  training,  casual  encounters,  and  the 
philosophical  sciences.  That  it  is  useful  as  a  training  is  obvious 
on  the  face  of  it.  The  possession  of  a  plan  of  inquiry  will 
enable  us  more  easily  to  argue  about  the  subject  proposed. 
For  purposes  of  casual  encounters,  it  is  useful  because  when 
we  have  counted  up  the  opinions  held  by  most  people,  we  shall 
meet  them  on  the  ground  not  of  other  people's  convictions 
but  of  their  own,  while  we  shift  the  ground  of  any  argument 
that  they  appear  to  us  to  state  unsoundly.  For  the  study  of 
the  philosophical  sciences  it  is  useful,  because  the  ability  to 
raise  searching  difficulties  on  both  sides  of  a  subject  will  make 
us  detect  more  easily  the  truth  and  arror  about  the  several 
points  that  arise.  It  has  a  further  use  in  relation  to  the 
ultimate  bases  of  the  principles  used  in  the  several  sciences. 
For  it  is  impossible  to  discuss  them  at  all  from  the  principles 
proper  to  the  particular  science  in  hand,  seeing  that  the  prin- 
ciples are  the  prius  of  everything  else :  it  is  through  the 
opinions  generally  held  on  the  particular  points  that  these 
have  to  be  discussed,  and  this  task  belongs  properly,  or  most 
appropriately,  to  dialectic:  for  dialectic  is  a  process  of  criti- 
cism wherein  lies  the  path  to  the  principles  of  all  inquiries. 

The  logic  thus  delineated  treats  of  propositions  and  problems, 
described  as  follows: 

11.02  The  materials  with  which  arguments  start  are  equal 
in  number,  and  are  identical,  with  the  subjects  on  which 
reasonings  take  place.  For  arguments  start  with  'propositions', 
while  the  subjects  on  which  reasonings  take  place  are  'pro- 

11.03  The  difference  between  a  problem  and  a  proposition 
is  a  difference  in  the  turn  of  the  phrase.  For  if  it  be  put  in 
this  way,  '  "An  animal  that  walks  on  two  feet"  is  the  definition 
of  man,  is  it  not  ?'  or  '  "Animal"  is  the  genus  of  man,  is  it  not  ?' 
the  result  is  a  proposition:  but  if  thus,  'Is  "an  animal  that 
walks  on  two  feet"  a  definition  of  man  or  no?'  the  result  is  a 
problem.  Similarly  too  in  other  cases.  Naturally,  then,  pro- 
blems and  propositions  are  equal  in  number:  for  out  of  every 
proposition  you  will  make  a  problem  if  you  change  the  turn 
of  the  phrase. 



Of  epoch-making  importance  is  the  classification  of  methods  of 
proof  given  in  the  same  connection : 

11.04  Having  drawn  these  definitions,  we  must  distinguish 
how  many  species  there  are  of  dialectical  arguments.  There 
is  on  the  one  hand  Induction,  on  the  other  Syllogism.  Now 
what  a  syllogism  is  has  been  said  before:  induction  is  a 
passage  from  individuals  to  universals,  e.g.  the  argument  that 
supposing  the  skilled  pilot  is  the  most  effective,  and  likewise 
the  skilled  charioteer,  then  in  general  the  skilled  man  is  the 
best  at  his  particular  task.  Induction  is  the  more  convinf ring 
and  clear:  it  is  more  readily  learnt  by  the  use  of  the  senses, 
and  is  applicable  generally  to  the  mass  of  men,  though 
syllogism  is  more  forcible  and  effective  against  contradictious 

The  subject  of  the  Topics  are  essentially  the  so-called  loci  (totcoi). 
Aristotle  never  defined  them,  and  so  far  no-one  has  succeeded  in 
saying  briefly  and  clearly  what  they  are.  In  any  case  it  is  a  matter 
of  certain  very  general  prescriptions  for  shaping  arguments. 

An  example: 

11.05  Now  one  commonplace  rule  (totzoc,)  is  to  look  and  see 
if  a  man  has  ascribed  as  an  accident  what  belongs  in  some  other 
way.  This  mistake  is  most  commonly  made  in  regard  to  the 
genera  of  things,  e.g.  if  one  were  to  say  that  white  happens  to 
be  a  colour  -  for  being  a  colour  does  not  happen  by  accident  to 
white,  but  colour  is  its  genus. 


As  an  introduction  to  these  loci  Aristotle  in  the  first  book  of  the 
Topics  developed  two  different  doctrines  of  the  structure  of  state- 
ments, both  of  which  obtained  considerable  historical  importance 
and  still  remain  of  interest:  namely  the  doctrines  of  the  so-called 
predicables  and  of  the  categories. 

11.06  Every  proposition  and  every  problem  indicates 
either  a  genus  or  a  peculiarity  or  an  accident  -  for  the  dif- 
ferentia too,  applying  as  it  does  to  a  class  (or  genus),  should 
be  ranked  together  with  the  genus.  Since,  however,  of  what 
is  peculiar  to  anything  part  signifies  its  essence,  while  part 
does  not,  let  us  divide  the  'peculiar'  into  both  the  aforesaid 



parts,  and  call  that  part  which  indicates  the  essence  a  'defi- 
nition', while  of  the  remainder  let  us  adopt  the  terminology 
which  is  generally  current  about  these  things,  and  speak  of 
it  as  a  'property'. 

11.07  We  must  now  say  what  are  'definition',  'property', 
'genus',  and  'accident'.  A  'definition'  is  a  phrase  signifying  a 
thing's  essence.  It  is  rendered  in  the  form  either  of  a  phrase 
in  lieu  of  a  term,  or  of  a  phrase  in  lieu  of  another  phrase;  for 
it  is  sometimes  possible  to  define  the  meaning  of  a  phrase  as 

11.08  A  'property'  (iSiov)  is  a  predicate  which  does  not 
indicate  the  essence  of  a  thing,  but  yet  belongs  to  that  thing 
alone,  and  is  predicated  convertibly  of  it.  Thus  it  is  a  property 
of  man  to  be  capable  of  learning  grammar:  for  if  A  be  a  man, 
then  he  is  capable  of  learning  grammar,  and  if  he  be  capable 
of  learning  grammar,  he  is  a  man. 

11.09  A  'genus'  is  what  is  predicated  in  the  category  of 
essence  of  a  number  of  things  exhibiting  differences  in  kind. 

11.10  An  'accident'  is  (1)  something  which  though  it  is 
none  of  the  foregoing  -  i.e.  neither  a  definition  nor  a  property 
nor  a  genus  -  yet  belongs  to  the  thing:  (2)  something  which 
may  possibly  either  belong  or  not  belong  to  any  one  and  the 
self-same  thing,  as  (e.g.)  the  'sitting  posture'  may  belong  or 
not  belong  to  some  self-same  thing. 

The  logical  significance  of  this  division  of  the  'predicates' 
consists  in  the  fact  that  it  is  an  attempt  to  analyse  propositions,  with 
reference  moreover  to  the  relation  between  subject  and  predicate. 
This  analysis  is  effected  in  terms  of  the  matter  rather  than  the  form, 
yet  contains  echoes  of  purely  structural  considerations,  as  for  instance 
in  the  distinction  between  genus  and  specific  difference  or  property, 
where  the  genus  is  evidently  symbolized  by  a  name,  properties  by  a 

As  a  kind  of  pendent  to  the  doctrine  of  the  predicables,  Aristotle 
presents  a  theory  of  identity: 

11.11  Sameness  would  be  generally  regarded  as  falling, 
roughly  speaking,  into  three  divisions.  We  generally  apply  the 
term  numerically  or  specifically  or  generically  -  numerically 
in  cases  where  there  is  more  than  one  name  but  only  one 
thing,  e.g.  'doublet'  and  'cloak';  specifically,  where  there  is 
more  than  one  thing,  but  they  present  no  differences  in  respect 
of  their  species,  as  one  man  and  another,   or  one  horse  and 



another:  for  things  like  this  that  fall  under  the  same  species 
are  said  to  be  'specifically  the  same'.  Similarly,  too,  those 
things  are  called  generically  the  same  which  fall  under  the 
same  genus,  such  as  a  horse  and  a  man. 


Another  analysis  of  propositions  is  contained  in  the  theory  of  the 
categories.  This  seems  to  be  a  systematic  development  of  hints  in 
Plato.  Only  in  one  place  (apart  from  the  Categories:  11.12)  do  we 
find  an  enumeration  of  ten  categories  (the  only  one  usually  ascribed 
to  Aristotle): 

11.13  Next,  then,  we  must  distinguish  the  classes  of 
predicates  in  which  the  four  orders  in  question  (11.06 — 11.10) 
are  found.  These  are  ten  in  number:  Essence,  Quantity,  Quali- 
ty, Relation,  Place,  Time,  Position,  State,  Activity,  Passivity. 
For  the  accident  and  genus  and  property  and  definition  of 
anything  will  always  be  in  one  of  these  categories :  for  all  the 
propositions  found  through  these  signify  either  something's 
essence  or  its  quality  or  quantity  or  some  one  of  the  other 
types  of  predicate.  It  is  clear,  too,  on  the  face  of  it  that  the 
man  who  signifies  something's  essence  signifies  sometimes  a 
substance,  sometimes  a  quality,  sometimes  some  one  of  the 
other  types  of  predicate.  For  when  a  man  is  set  before  him 
and  he  says  that  what  is  set  there  is  'a  man'  or  'an  animal', 
he  states  its  essence  and  signifies  a  substance;  but  when  a 
white  colour  is  set  before  him  and  he  says  that  what  is  set 
there  is  'white'  or  is  'a  colour',  he  states  its  essence  and  signi- 
fies a  quality.  Likewise,  also,  if  a  magnitude  of  a  cubit  be 
set  before  him  and  he  says  that  what  is  set  there  is  a  magnitude 
of  a  cubit,  he  will  be  describing  its  essence  and  signifying  a 
quantity.  Likewise  also  in  the  other  cases. 

This  text  contains  an  ambiguity:  'essence'  (xi  kaxi)  means  first 
a  particular  category  -  that  of  substance  (oucria)  as  we  see  from  a 
parallel  text  of  the  Categories  (11.14)  -  secondly  that  essence  or 
intrinsic  nature  which  is  found  in  every  category,  not  only  in  that  of 
substance.  The  thought  becomes  clear  if  'substance'  is  put  in  the 
list  of  the  ten  categories  in  place  of  'essence'. 

Here  the  doctrine  of  the  categories  is  treated  as  a  division  of 
sentences  and  problems  for  practical  purposes.  But  beyond  this 
Aristotle  regarded  it  as  involving  two  more  important  problems. 
In  the  Prior  Analytics  we  read : 



11.15  The  expressions  'this  belongs  to  that'  and  'this  holds 
true  of  that'  must  be  understood  in  as  many  ways  as  there 
are  different  categories. 

That  means  that  the  so-called  copula  of  the  sentence  has  as  many 
meanings  as  there  are  categories.  That  is  the  first  reason  why  the 
theory  of  the  categories  is  logically  so  important.  The  second  is 
that  while  this  theory  constitutes  an  attempt  at  classifying  objects 
according  to  the  ways  in  which  they  are  predicable,  it  put  in  Aristo- 
tle's path  the  problem  of  the  univeral  class.  He  solved  it  with 
brilliant  intuition,  though,  as  we  now  know,  with  the  help  of  a 
faulty  proof.  The  relevant  passage  occurs  in  the  third  book  of  the 

11.16  It  is  not  possible  that  either  unity  or  being  should 
be  a  single  genus  of  things;  for  the  differentiae  of  any  genus 
must  each  of  them  both  have  being  and  be  one,  but  it  is  not 
possible  for  the  genus  taken  apart  from  its  species  (any  more 
than  for  the  species  of  the  genus)  to  be  predicated  of  its  proper 
differentiae;  so  that  if  unity  or  being  is  a  genus,  no  differentia 
will  either  have  being  or  be  one. 

The  line  of  thought  which  Aristotle  expresses  in  this  very  com- 
pressed formula  is  as  follows: 

1.  For  all  A:  if  A  is  a  genus,  then  there  is  (at  least)  one  B,  which 
is  the  specific  difference  of  A; 

2.  for  all  A  and  B:  if  B  is  the  specific  difference  of  A,  then  not: 
B  is  A.  Suppose  now 

3.  there  is  an  all-inclusive  genus  V:  of  this  it  would  be  true  that 

4.  for  every  B:  B  is  V. 

As  V  is  a  genus,  it  must  have  a  difference  (by  1.);  call  it  D.  Of  this 
D  it  would  be  true  on  the  one  hand  that  D  is  V  (by  4.),  and  on  the 
other  that  D  is  not  V  (by  2.).  Thus  a  contradiction  results,  and 
at  least  one  of  the  premisses  must  be  false  (cf.  16.33).  As  Aristotle 
holds  1.  and  2.  to  be  true,  he  must  therefore  reject  the  supposition 
that  there  is  an  all-inclusive  genus  (3.):  there  is  no  summum  genus. 
We  have  here  the  basis  of  the  scholastic  doctrine  of  analogy  (28.19) 
and  the  first  germ  of  a  theory  of  types  (cf.  §  47). 

The  proof  is  faulty :  for  'D  is  V  is  not  false  but  meaningless  (48.24). 
But  beyond  all  doubt  the  thought  confronting  us  deserves  to  be 
styled  a  brilliant  intuition. 


The  last  book  of  the  Topics,  known  as  the  Sophistic  Befutations, 
contains  an  extensive  doctrine  of  fallacious  inferences.  Like  most 
other  parts  of  the  Topics  this  one  too  belongs  to  the  first  form  of 
Aristotelian   logic,   not  yet   formal,    but   guided    by   the   practical 



interests  of  every-day  discussion.  There  is  a  second  doctrine  of 
fallacious  inference,  in  the  Prior  Analytics  (11.17j,  much  briefer 
than  the  first  but  incomparably  more  formal;  all  fallacious  inferences 
are  there  reduced  to  breaches  of  syllogistic  laws.  However  neither 
Aristotle  himself,  nor  anyone  after  him,  really  succeeded  in  replacing 
the  doctrine  of  the  Sophistic  Refutations,  primitive  though  it  is 
from  the  formal  point  of  view.  Knowledge  of  it  is  also  indispensable 
for  the  understanding  of  scholastic  logic.  For  all  of  which  reasons 
we  shall  cite  a  few  passages  from  it  here. 

11.18  Refutation  is  reasoning  involving  the  contradictory 
of  the  given  conclusion.  Now  some  of  them  do  not  really 
achieve  this,  though  they  seem  to  do  so  for  a  number  of 
reasons;  and  of  these  the  most  prolific  and  usual  domain  is 
the  argument  that  turns  upon  names  only.  It  is  impossible  in 
a  discussion  to  bring  in  the  actual  things  discussed :  we  use 
their  names  as  symbols  instead  of  them;  and  therefore  we 
suppose  that  what  follows  in  the  names,  follows  in  the  things 
as  well,  just  as  people  who  calculate  suppose  in  regard  to 
their  counters.  But  the  two  cases  (names  and  things)  are  not 
alike.  For  names  are  finite  and  so  is  the  sum-total  of  formulae, 
while  things  are  infinite  in  number.  Inevitably,  then,  the  same 
formulae,  and  a  single  name,  have  a  number  of  meanings. 

Historically,  a  very  important  text:  in  it  Aristotle  rejects  forma- 
lism, rightly  so  for  the  purposes  of  ordinary  language.  For  without 
preliminary  distinction  of  the  various  functioning  of  signs  correct 
laws  cannot  be  formulated  in  such  a  language.  The  text  just  cited 
underlies  the  vast  growth  of  medieval  doctrine  about  supposition, 
appellation  and  analogy  (§§  27  and  28).  So  far  as  concerns  Aristotle 
and  the  other  ancient  logicians,  it  appears  that  they  got  round  the 
difficulty  mentioned  by  applying  rules  by  which  ordinary  language 
was  turned  into  an  artificial  language  with  a  single  function  for 
every  verbal  form. 

11.19  There  are  two  styles  of  refutation:  for  some  depend 
on  the  language  used,  while  some  are  independent  of  language. 
Those  ways  of  producing  the  false  appearance  of  an  argument 
which  depend  on  language  are  six  in  number:  they  are 
ambiguity,  amphiboly,  combination,  division  of  words, 
accent,  form  of  expression. 

11.20  Arguments  such  as  the  following  depend  upon  ambi- 
guity. 'Those  learn  who  know:  for  it  is  those  who  know  their 
letters  who  learn  the  letters  dictated  to  them.'  For  'to  learn' 



is  ambiguous;  it  signifies  both   'to  understand'   by  the  use 
of  knowledge,  and  also  'to  acquire  knowledge'. 

11.21  Examples  such  as  the  following  depend  upon  amphi- 
boly: .  .  .  'Speaking  of  the  silent  is  possible':  for  'speaking  of 
the  silent'  also  has  a  double  meaning:  it  may  mean  that  the 
speaker  is  silent  or  that  the  things  of  which  he  speaks  are  so. 

11.22  Upon  the  combination  of  words  there  depend 
instances  such  as  the  following:  'A  man  can  walk  while 
sitting,  and  can  write  while  not  writing'.  For  the  meaning  is 
not  the  same  if  one  divides  the  words  and  if  one  combines 
them  in  saying  that  'it  is  possible  to  walk-while-sitting  .  .  .'. 
The  same  applies  to  the  latter  phrase,  too,  if  one  combines  the 
words  'to  write-while-not-writing' :  for  then  it  means  that  he 
has  the  power  to  write  and  not  to  write  at  once;  whereas  if 
one  does  not  combine  them,  it  means  that  when  he  is  not 
writing  he  has  the  power  to  write. 

11.23  Upon  division  depend  the  propositions  that  5  is  2 
and  3,  and  even  and  odd,  and  that  the  greater  is  equal:  for 
it  is  that  amount  and  more  besides. 

11.24  Of  fallacies,  on  the  other  hand,  that  are  independent 
of  language  there  are  seven  kinds : 

(1)  that  which  depends  upon  Accident: 

(2)  the  use  of  an  expression  absolutely  or  not  absolutely  but 
with  some  qualification  of  respect,  or  place,  or  time,  or 
relation : 

(3)  that  which  depends  upon  ignorance  of  what  'refutation' 

(4)  that  which  depends  upon  the  consequent: 

(5)  that  which  depends  upon  assuming  the  original  con- 
clusion : 

(6)  stating  as  cause  what  is  not  the  cause: 

(7)  the  making  of  more  than  one  question  into  one. 

And  example  of  (1)  is:  'If  Coriscus  is  different  from  a  man  he  is 
different  from  himself  (11.25);  of  (2):  'Suppose  an  Indian  to  be 
black  all  over,  but  white  in  respect  of  his  teeth ;  then  he  is  both  white 
and  not  white'  (11.26);  (3)  consists  in  proving  something  other 
than  what  is  to  be  proved  (11.27);  (5)  consists  in  presupposing  what 
is  to  be  proved  (11.28).  (4)  alone  involves  a  formal  fallacy,  namely 
concluding  from  the  consequent  to  the  antecedent  of  a  conditional 
sentence  (11.29). 




Aristotle  developed  two  different  theories  of  opposition.  The 
first,  contained  in  the  Topics  (12.01)  and  belonging  to  the  earlier 
period  of  his  development,  is  most  clearly  summarized  in  the 
pseudo-aristotelian  Categories : 

12.02  There  are  four  senses  in  which  one  thing  is  said  to  be 
opposed  to  another:  as  correlatives,  or  as  contraries,  or  as 
privation  and  habit  (e?i?),  or  as  affirmation  and  denial.  To 
give  a  general  outline  of  these  oppositions:  the  double  is 
correlative  to  the  half,  evil  is  contrary  to  good,  blindness  is  a 
privation  and  sight  a  habit,  'he  sits'  is  an  affirmation,  'he  does 
not  sit'  a  denial. 

Two  points  are  worth  remark:  the  division  presupposes  a  material 
standpoint,  and  even  in  the  last  case,  contradictory  opposition, 
concerns  relationships  between  terms,  not  sentences. 

It  is  quite  otherwise  in  the  later  period.  The  second  doctrine 
presupposes  the  Aristotelian  theory  of  quantification,  which  is  later 
than  the  Topics: 

12.03  Some  things  are  universal,  others  individual.  By  the 
term  'universal'  I  mean  that  which  is  of  such  a  nature  as  to  be 
predicated  of  many  subjects,  by  'individual'  that  which  is  not 
thus  predicated.  Thus  'man'  is  a  universal,  'Callias'  an  indi- 
vidual. .  .  .  If, N  then,  a  man  states  a  positive  and  a  negative 
proposition  of  universal  character  with  regard  to  a  universal, 
these  two  propositions  are  'contrary'.  By  the  expression  'a 
proposition  of  universal  character  with  regard  to  a  universal', 
such  propositions  as  'every  man  is  white',  'no  man  is  white' 
are  meant.  When,  on  the  other  hand,  the  positive  and  negative 
propositions,  though  they  have  regard  to  a  universal,  are  yet 
not  of  universal  character,  they  will  not  be  contrary,  albeit  the 
meaning  intended  is  sometimes  contrary.  As  instances  of 
propositions  made  with  regard  to  a  universal,  but  not  of 
universal  character,  we  may  take  the  propositions  'man  is 
white',  'man  is  not  white'.  'Man  'is  a  universal,  but  the  pro- 
position is  not  made  as  of  universal  character;  for  the  word 
'every'  does  not  make  the  subject  a  universal,  but  rather 
gives  the  proposition  a  universal  character.  If,  however,  both 



predicate  and  subject  are  distributed,  the  proposition  thus 
constituted  is  contrary  to  truth;  no  affirmation  will,  under 
such  circumstances,  be  true.  The  proposition  'every  man  is 
every  animal'  is  an  example  of  this  type. 

This  text  contains  the  following  points  of  doctrine:  1.  distinction 
between  general  and  singular  sentences,  according  to  the  kind  of 
subject;  2.  divison  of  general  sentences  into  universal  and  particular 
according  to  the  extension  of  the  subject;  3.  rejection  of  quantifica- 
tion of  the  predicate.  The  whole  doctrine  is  now  purely  formal,  and 
is  explicitly  concerned  with  sentences. 

Another  division  is  to  be  found  at  the  beginning  of  the  Prior 

12.04  A  premiss  then  is  a  sentence  affirming  or  denying  one 
thing  of  another.  This  is  either  universal  or  particular  or 
indefinite.  By  universal  I  mean  the  statement  that  something 
belongs  to  all  or  none  of  something  else;  by  particular  that  it 
belongs  to  some  or  not  to  some  or  not  to  all;  by  indefinite  that 
it  does  or  does  not  belong,  without  any  mark  to  show  whether 
it  is  universal  or  particular. 

Here  Aristotle  enumerates  three  kinds  of  sentence:  universal, 
particular  and  indefinite.  It  is  striking  that  no  mention  is  made  of 
singular  sentences.  This  is  due  to  the  fact  that  every  term  in  the 
syllogistic  must  be  available  both  as  subject  and  predicate,  but  accord- 
ing to  Aristotle  singular  terms  cannot  be  predicated  (12.05).  In  the 
particular  sentence  'some'  means  'at  least  one,  not  excluding  all'. 
Whereas,  as  Sugihara  has  recently  shewn  (12.06),  an  indefinite 
sentence  should  probably  be  interpreted  in  the  sense:  'at  least  one 
A  is  B  and  at  least  one  A  is  not  B\  However  cases  in  which  the 
formal  properties  of  particular  and  indefinite  sentences  differ  are 
rare  in  the  syllogistic,  so  that  Aristotle  himself  often  states  the 
equivalence  of  these  sentences  (12.07).  Later,  even  in  Alexander  of 
Aphrodisias  (12.08),  these  cases  are  dropped  altogether. 

12.09  Verbally  four  kinds  of  opposition  are  possible,  viz. 
universal  affirmative  to  universal  negative,  universal  affirma- 
tive to  particular  negative,  particular  affirmative  to  universal 
negative,  and  particular  affirmative  to  particular  negative: 
but  really  there  are  only  three :  for  the  particular  affirmative  is 
only  verbally  opposed  to  the  particular  negative.  Of  the 
genuine  opposites  I  call  those  which  are  universal  contraries, 
the  universal  affirmative  and  the  universal  negative,  e.g.  'all 
science  is  good',  'no  science  is  good';  the  others  I  call  contra- 



Here  we  have  the  'logical  square'  later  to  become  classical,  which 
can  be  set  out  schematically  thus: 


to  belong  to  all  contrary  to  belong  to  none 



%    .if* 


/     * . 
/         \ 

to  belong  to  some  only  verbal  not  to  belong  to  all 

The  logical  relationships  here  intended  are  shown  in  the  following 

12.10  Of  such  corresponding  positive  and  negative  pro- 
positions as  refer  to  universals  and  have  a  universal  character, 
one  must  be  true  and  the  other  false. 

12.11  It  is  evident  also  that  the  denial  corresponding  to  a 
single  affirmation  is  itself  single;  ...  for  instance,  the  affir- 
mation 'Socrates  is  white'  has  its  proper  denial  in  the  pro- 
position 'Socrates  is  not  white'  .  .  .  The  denial  proper  to  the 
affirmation  'every  man  is  white'  is  'not  every  man  is  white'; 
that  proper  to  the  affirmation  'some  man  is  white'  is  'no  man 
is  white'. 

In  the  later  tradition  the  so-called  laws  of  subalternation  also 
came  to  be  included  in  the  'logical  square'.  They  run: 
If  A  belongs  to  all  B,  then  it  belongs  to  some  B  (12.12). 
If  A  belongs  to  no  B,  then  to  some  B  it  does  not  belong  (12.13). 


In  Aristotle's  logic  negation  normally  occurs  only  as  a  functor 
determining  a  sentence,  but  there  are  a  number  of  places  in  the 
Organon  where  formulae  are  considered  which  contain  a  negation 
determining  a  name.  Thus  we  read  in  the  Hermeneia: 

12.14  The  proposition  'no  man  is  just'  follows  from  the 
proposition  'every  man  is  not-just'  and  the  proposition  'not 
every  man  is  not-just',  which  is  the  contradictory  of  'every 



man  is  not-just',  follows  from  the  proposition  'some  man  is 
just';  for  if  this  be  true,  there  must  be  some  just  man. 

These  laws  were  evidently  discovered  with  great  labour  and  after 
experimenting  with  various  false  formulae  (12.15).  Aristotle  has 
similar  ones  for  sentences  with  individual  subjects  in  the  Hermeneia 

In  the  Prior  Analytics  Aristotle  develops  a  similar  doctrine  in 
more  systematic  form  and  with  variables: 

12.17  Let  A  stand  for  'to  be  good',  B  for  'not  to  be  good', 
let  C  stand  for  'to  be  not-good',  and  be  placed  under  B,  and 
let  D  stand  for  'not  to  be  not-good'  and  be  placed  under  A. 
Then  either  A  or  B  will  belong  to  everything,  but  they  will 
never  belong  to  the  same  thing;  and  either  C  or  D  will  belong 
to  everything,  but  they  will  never  belong  to  the  same  thing. 
And  B  must  belong  to  everything  to  which  C  belongs.  For  if 
it  is  true  to  say  'it  is  not-white',  it  is  true  also  to  say  'it  is  not 
white' :  for  it  is  impossible  that  a  thing  should  simultaneously 
be  white  and  be  not-white,  or  be  a  not-white  log  and  be  a 
white  log;  consequently  if  the  affirmation  does  not  belong, 
the  denial  must  belong.  But  C  does  not  always  belong  to  B: 
for  what  is  not  a  log  at  all,  cannot  be  a  not-white  log  either. 
On  the  other  hand  D  belongs  to  everything  to  which  A  belongs. 
For  either  C  or  D  belongs  to  everything  to  which  A  belongs. 
But  since  a  thing  cannot  be  simultaneously  not-white  and 
white,  D  must  belong  to  everything  to  which  A  belongs.  For 
of  that  which  is  white  it  is  true  to  say  that  it  is  not  not-white. 
But  A  is  not  true  of  all  D.  For  of  that  which  is  not  a  log  at  all 
it  is  not  true  to  say  A,  viz.  that  it  is  a  white  log.  Consequently 
D  is  true,  but  A  is  not  true,  i.e.  that  it  is  a  white  log.  It  is  clear 
also  that  A  and  C  cannot  together  belong  to  the  same  thing, 
and  that  B  and  D  may  possibly  belong  to  the  same  thing. 


While  Aristotle  was  well  acquainted  with  the  principle  of  identity, 
so  much  discussed  later,  he  only  mentions  it  in  passing  (12.18).  But 
to  the  principle  of  contradiction  he  devoted  the  whole  of  Book  T 
of  the  Metaphysics.  This  book  is  evidently  a  youthful  work,  and  was 
perhaps  written  in  a  state  of  excitement,  since  it  contains  logical 
errors;  nevertheless  it  is  concerned  with  an  intuition  of  fundamental 
importance  for  logic. 



The  following  are  the  most  important  formulations  of  the  principle 
of  contradiction: 

12.19  The  same  attribute  cannot  at  the  same  time  belong 
and  not  belong  to  the  same  subject  and  in  the  same  respect. 

12.20  Let  A  stand  for  'to  be  good',  B  for  'not  to  be  good' .... 
Then  either  A  or  B  will  belong  to  everything,  but  they  will 
never  belong  to  the  same  thing. 

12.21  It  is  impossible  that  contradictories  should  be  at  the 
same  time  true  of  the  same  thing. 

12.22  It  is  impossible  to  affirm  and  deny  truly  at  the  same 

The  first  two  of  these  formulae  are  in  the  object  language,  the 
last  two  in  a  metalanguage,  and  the  author  evidently  understands 
the  difference. 

In  the  Topics  and  Hermeneia  Aristotle  has  a  stronger  law: 

12.23  It  is  impossible  that  contrary  predicates  should  belong 
at  the  same  time  to  the  same  thing. 

12.24  Propositions  are  opposed  as  contraries  when  both 
the  affirmation  and  the  denial  are  universal  ...  in  a  pair  of 
this  sort  both  propositions  cannot  be  true. 

This  last  statement  is  quite  understandable  if  one  remembers 
that  when  Aristotle  was  young  proofs  principally  consisted  of 
refutations.  But  when  Aristotle  had  developed  his  syllogistic,  in 
which  refutation  has  only  a  subordinate  part  to  play,  he  not  only 
found  that  the  principle  of  contradiction  would  not  do  at  all  as  the 
first  axiom,  but  also  that  violence  may  be  done  to  it  in  a  correct 

The  first  in  modern  times  to  advert  to  this  Aristotelian  doctrine 
was  I.  Husic  in  1906  (12.27).  It  may  seem  so  astonishing  to  readers 
accustomed  to  the  'classical'  interpretation  of  Aristotelian  logic, 
that  it  is  worth  while  shewing  not  merely  its  absolute  necessity 
but  also  the  context  from  which  our  logician's  thought  clearly 

12.28  The  law  that  it  is  impossible  to  affirm  and  deny 
simultaneously  the  same  predicate  of  the  same  subject  is  not 
expressly  posited  by  any  demonstration  except  when  the 
conclusion  also  has  to  be  expressed  in  that  form;  in  which  case 
the  proof  lays  down  as  its  major  premiss  that  the  major 
is  truly  affirmed  of  the  middle  but  falsely  denied.  It  makes  no 
difference,  however,  if  we  add  to  the  middle,  or  again  to  the 



minor  term,  the  corresponding  negative.  For  grant  a  minor 
term  of  which  it  is  true  to  predicate  man  -  even  if  it  be  also 
true  to  predicate  not-man  of  it  -still  grant  simply  that  man  is 
animal  and  not  not-animal,  and  the  conclusion  follows:  for 
it  will  still  be  true  to  say  that  Callias  -  even  if  it  be  also  true 
to  say  that  not-Callias  -  is  animal  and  not  not-animal. 

The  syllogism  here  employed  has  then,  omitting  quantifiers, 
the  following  form: 

12.281  If  M  is  P  and  not  not-P,  and  £  is  M,  then  S  is  P 
and  not  not-P. 

So  the  principle  of  contradiction  is  no  axiom,  and  does  not  need 
to  be  presupposed,  except  in  syllogisms  of  the  fore-going  kind. 
The  text  quoted  is  also  remarkable  in  that  the  middle  term  in  12.281 
is  a  product  (cf.  the  commentary  on  15.151),  and  that  in  the  minor 
term  an  individual  name  is  substituted  (cf.  §  13,  C,  5)  -  in  each 
case  contrary  to  normal  syllogistic  practice.  However  the  text 
comes  from  the  Posterior  Analytics  and  must  belong  to  a  relatively 
early  period. 

Aristotle  goes  still  further  and  states  that  the  principle  of  con- 
tradiction can  be  completely  violated  in  a  conclusive  syllogism: 

12.29  In  the  middle  (i.e.  second:  cf.  13.07)  figure  a  syllo- 
gism can  be  made  both  of  contradictories  and  contraries.  Let 
'A'  stand  for  'good',  let  ' B'  and  'C  stand  for  'science'.  If  then 
one  assumes  that  every  science  is  good,  and  no  science  is  good, 
A  belongs  to  all  B  and  to  no  C,  so  that  B  belongs  to  no  C: 
no  science  is  then  a  science. 

This  syllogism  has  the  following  form : 

12.291   If  all  M  is  P  and  no  M  is  P,  then  no  M  is  M. 


One  formulation  of  this  principle  has  already  been  quoted  (12.20). 
Others  are: 

12.30  In  the  case  of  that  which  is,  or  which  has  taken  place, 
propositions,  whether  positive  or  negative,  must  be  true  or 
false.  Again,  in  the  case  of  a  pair  of  contradictories,  either  when 
the  subject  is  universal  and  the  propositions  are  of  a  universal 
character,  or  when  it  is  individual,  as  has  been  said,  one  of  the 
two  must  be  true  and  the  other  false. 



12.31  One  side  of  the  contradiction  must  be  true.  Again,  if 
it  is  necessary  with  regard  to  everything  either  to  assert  or  to 
deny  it,  it  is  impossible  that  both  should  be  false. 

Aristotle's  normal  practice  was  to  presuppose  the  correctness  of 
these  theses,  and  he  devoted  a  notable  chapter  of  the  fourth  hook  of 
the  Metaphysics  (T  8)  to  the  defence  of  the  principle  of  tertium 
exclusion  (or  tertium  non  dalur).  At  least  once,  however,  he  called  it 
into  question:  in  the  ninth  chapter  of  the  Hermeneia  he  will  not 
allow  it  to  be  valid  for  future  contingent  events.  He  bases  his  ar- 
gument thus: 

12.32  If  it  is  true  to  say  that  a  thing  is  white,  it  must 
necessarily  be  white;  if  the  reverse  proposition  is  true,  it 
will  of  necessity  not  be  white  ....  It  may  therefore  be  argued 
that  it  is  necessary  that  affirmations  or  denials  must  be 
either  true  or  false.  Now  if  this  be  so,  nothing  is  or  takes 
place  fortuitously,  either  in  the  present  or  in  the  future,  and 
there  are  no  real  alternatives;  everthing  takes  place  of  neces- 
sity and  is  fixed.  ...  It  is  therefore  plain  that  it  is  not  necessary 
that  of  an  affirmation  and  a  denial  one  should  be  true  and  the 
other  false.  For  in  the  case  of  that  which  exists  potentially, 
but  not  actually,  the  rule  which  applies  to  that  which  exists 
actually  does  not  hold  good. 

These  considerations  had  no  influence  on  Aristotle's  logical 
system,  as  has  already  been  said,  but  they  came  to  have  great 
historical  importance  in  the  Middle  Ages. 

The  doubt  about  the  validity  ot  the  principle  of  tertium  exclusum 
arose  from  an  intuition  of  the  difficult  problems  which  it  sets.  The 
debate  is  not  closed  even  today. 


We  give  here  a  page  of  the  Prior  Analytics  in  as  literal  a  trans- 
lation as  possible,  and  comment  on  it  afterwards.  It  contains  the 
essentials  of  what  later  came  to  be  called  Aristotle's  assertoric 
syllogistic.  It  is  a  leading  text,  in  which  no  less  than  three  great 
discoveries  are  applied  for  the  first  time  in  history:  variables,  purely 
formal  treatment,  and  an  axiomatic  system.  It  constitutes  the 
beginning  of  formal  logic.  Short  as  it  is,  it  formed  the  basis  of  logical 
speculation  for  more  than  two  thousand  years  -  and  yet  has  been 
only  too  often  much  misunderstood.  It  deserves  to  be  read  atten- 



Aristotle  begins  by  stating  the  laws  of  conversion  of  sentences. 
These  are  cited  below,  14.10  ff.,  among  the  bases  of  the  systematic 
development.  He  goes  on : 

13.01  When  then  three  terms  are  so  related  one  to  another 
that  the  last  is  in  the  middle  (as  in  a)  whole  and  the  middle  is  or  is 
not  in  the  first  as  in  a  whole,  then  there  must  be  a  perfect 
syllogism  of  the  two  extremes. 

13.02  Since  if  A  (is  predicated)  of  all  B,  and  B  of  all  C,  A 
must  be  predicated  of  all  C. 

13.03  Similarly  too  if  A  (is  predicated)  of  no  B,  and  B  of 
all  A,  it  is  necessary  that  A  will  belong  to  no  C. 

13.04  But  if  the  first  follows  on  all  the  middle  whereas  the 
middle  belongs  to  none  of  the  last,  there  is  no  syllogism  of  the 
extremes;  for  nothing  necessary  results  from  these;  for  the 
first  may  belong  to  all  and  to  none  of  the  last;  so  that  neither 
a  particular  nor  a  universal  is  necessary;  and  since  there  is 
nothing  necessary  these  produce  no  syllogism.  Terms  for 
belonging  to  all:  animal,  man,  horse;  for  belonging  to  none: 
animal,  man,  stone. 

13.05  But  if  one  of  the  terms  is  related  wholly,  one  partially, 
to  the  remaining  one;  when  the  wholly  related  one  is  posited 
either  affirmatively  or  negatively  to  the  major  extreme,  and 
the  partially  related  one  affirmatively  to  the  minor  extreme, 
there  must  be  a  perfect  syllogism  .  .  .  for  let  A  belong  to  all  B 
and  B  to  some  C,  then  if  being  predicated  of  all  is  what  has 
been  said,  A  must  belong  to  some  C. 

13.06  And  if  A  belongs  to  no  B  and  B  to  some  C,  to  some 
C  A  must  not  belong  .  .  .  and  similarly  if  the  BC  (premiss)  is 
indefinite  and  affirmative. 

13.07  But  when  the  some  belongs  to  all  of  one,  to  none  of 
the  other;  such  a  figure  I  call  the  second. 

13.08  For  let  M  be  predicated  of  no  N  and  of  all  X ;  since 
then  the  negative  converts,  N  will  belong  to  no  M ;  but  M 
was  assumed  (to  belong)  to  all  X ;  so  that  N  (will  belong)  to 
no  X ;  for  this  has  been  shewn  above. 

13.09  Again,  if  M  (belongs)  to  all  iV  and  to  no  X,  X  too 
will  belong  to  no  N;  for  if  M  (belongs)  to  no  X,  X  too  (belongs) 
to  no  M ;  but  M  belonged  to  all  N ;  therefore  X  will  belong  to 
no  iV,  for  the  first  figure  has  arisen  again ;  but  since  the  nega- 
tive converts,  N  too  will  belong  to  no  X,  so  that  it  will  be  the 



same  syllogism.    It  is  possible  to  shew  this  by  bringing  to 

13.10  If  M  belongs  to  no  TV  and  to  some  X,  then  to  some 
X  TV  must  not  belong.  For  since  the  negative  converts,  TV 
will  belong  to  no  M,  but  M  has  been  supposed  to  belong  to 
some  X;  so  that  to  some  X  TV  will  not  belong;  for  a  syllogism 
arises  in  the  first  figure. 

13.11  Again,  if  M  belongs  to  all  TV  and  to  some  X  not.  to 
some  TV  X  must  not  belong.  For  if  it  belongs  to  all,  and  M  is 
predicated  of  all  TV,  M  must  belong  to  all  X ;  but  to  some  it  has 
been  supposed  not  to  belong. 

13.12  If  to  the  same,  one  belongs  to  all,  the  other  to  none,  or 
both  to  all  or  to  none,  such  a  figure  I  call  third. 

13.13  When  both  P  and  R  belong  to  all  S,  of  necessity  P 
will  belong  to  some  B;  for  since  the  affirmative  converts,  S 
will  belong  to  some  P,  so  that  when  P  (belongs)  to  all  S,  and 
S  to  some  R,  P  must  belong  to  some  R;  for  a  syllogism 
arises  in  the  first  figure.  One  can  make  the  proof  also  by  (bring- 
ing to)  the  impossible  and  by  setting  out  (terms);  for  if  both 
belong  to  all  S,  if  some  of  the  (things  which  are)  S  be  taken,  say  TV, 
to  this  both  P  and  R  will  belong,  so  that  to  some  R  P  will  belong. 

13.14  And  if  R  belongs  to  all  S,  and  P  to  none,  there  will 
be  a  syllogism  that  to  some  R  P  necessarily  will  not  belong. 
For  there  is  the  same  manner  of  proof,  with  the  RS  premiss 
converted.  It  could  also  be  shewn  by  the  impossible  as  in  the 
previous  cases. 

13.15  If  R  (belongs)  to  all  S  and  P  to  some,  P  must  belong 
to  some  R.  For  since  the  affirmative  converts,  8  will  belong 
to  some  P,  so  that  when  R  (belongs)  to  all  S,  and  S  to  some  P, 
R  too  will  belong  to  some  P;  so  that  P  (will  belong)  to  some  R. 

13.16  Again,  if  R  belongs  to  some  S  and  P  to  all,  P  must 
belong  to  some  R;  for  there  is  the  same  manner  of  proof. 
One  can  also  prove  it  by  the  impossible  and  by  setting  out, 
as  in  the  previous  cases. 

13.17  If  R  belongs  to  all  S,  and  to  some  (S)  P  does  not. 
then  to  some  R  P  must  not  belong.  For  if  to  all,  and  R  (belongs) 
to  all  S,  P  will  also  belong  to  all  S;  but  it  did  not  belong.  It  is 
also  proved  with  reduction  (to  the  impossible)  if  some  of 
what  is  S  be  taken  to  which  P  does  not  belong. 

13.18  If  P  belongs  to  no  S,  and  R  to  some  S,  to  some  R  P 
will  not  belong;  for  again  there  will  be  the  first  figure  when  the 
RS  premiss  is  converted. 



This  passage  is  composed  in  such  compressed  language  that  most 
readers  find  it  very  hard  to  understand.  Indeed  the  very  style  is  of 
the  greatest  significance  for  the  history  of  logic ;  for  here  we  have  the 
manner  of  thought  and  writing  of  all  genuine  formal  logicians,  be 
they  Stoics  or  Scholastics,  be  their  name  Leibniz  or  Frege.  Hence 
we  have  given  a  literal  version,  but  shall  now  interpret  it  with  the 
aid  of  paraphrase  and  commentary: 

on  13.01 :  Aristotle  defines  the  first  figure.  This  may  serve  as  an 
example:  Gainful  art  is  contained  in  art  in  general  as  in  a  whole; 
the  art  of  pursuit  (e.g.  hunting)  is  contained  in  gainful  art  as  in 
a  whole;  therefore  the  art  of  pursuit  is  contained  in  art  in  general 
as  in  a  whole.  The  example  is  taken  from  Plato's  division  (8.05), 
from  which  the  Aristotelian  syllogism  seems  to  have  developed. 

We  shall  explain  what  a  perfect  syllogism  is  in  §  14. 

on  13.02:  This  mood  later  (with  Peter  of  Spain)  came  to  be  called 
'Barbara'.  Hereafter  we  refer  to  all  moods  by  the  mnemonic  names 
originating     with     Peter     of     Spain     (cf.     32.04  ff.). 

We  obtain  an  example  by  substitution: 
If  animal  belongs  to  all  man 
and  man  belongs  to  all  Greek, 
then  animal  belongs  to  all  Greek. 

on  13.03:  Celarent:   If  stone  belongs  to  no  man 

and  man  belongs  to  all  Greek, 
then  stone  belongs  to  no  Greek. 

on  13.04:  Names  are  here  given  for  two  substitutions  by  which 
it  can  be  shewn  that  a  further  mood  is  invalid.  Probably  the  follow- 
ing are  intended : 

Mood  Substitution 

If  A  to  all  B  1.   If  animal  belongs  to  all  man 

and  B  to  no  C,  and  man  belongs  to  no  horse, 

then  A  to  no  C,  then  animal  belongs  to  no  horse. 

2.   If  animal  belongs  to  all  man 
and  man  belongs  to  no  stone, 
then  animal  belongs  to  no  stone. 
In  each  case  the  premisses  are  true,  but  the  conclusion  is  once 
true,  once  false.  Therefore  the  mood  is  invalid. 

on  13.05:  Darii:     If  Greek  belongs  to  all  Athenian 

and   Athenian   belongs   to   some   logician, 
then  Greek  belongs  to  some  logician. 
It  is  to  be  noticed  that  here  and  subsequently  'some'  must  have 
the  sense  of  'at  least  one'. 

on  13.06:  Ferio:     If  Egyptian  belongs  to  no  Greek 

and  Greek  belongs  to  some  logician, 

then  to  some  logician  Egyptian  does  not  belong. 



on  13.07:  Aristotle  defines  the  second  figure,  in  which  the  middle 
term  is  predicate  in  both  premisses.  He  considers  three  cases:  1.  one 
premiss  is  universal  and  affirmative,  the  other  universal  and  negative, 
2.  both  premisses  are  universal  and  affirmative,  '.>.  both  premisses 

are  negative.  Only  in  the  first  case  there  are  valid  syllogisms, 
on  13.08:  Cesare:    If  man  belongs  to  no  stone 

and  man  belongs  to  all  Greek, 
then  stone  belongs  to  no  Greek. 
This  is  reduced  to  Celarent  (13.03)  by  conversion  of  the  major 
(first)  premiss:  If  stone  belongs  to  no  man 

and  man  belongs  to  all  Greek, 
then  stone  belongs  to  no  Greek, 
on  13.09:  Camestres:    If  (1)  animal  belongs  to  all  man 

and  (2)  animal  belongs  to  no  stone, 
then  (3)  man  belongs  to  no  stone. 
The  proof  proceeds  by  reduction  to  Celarent  (13.03).  First  the  minor 
premiss  (2)  is  converted : 

(4)  stone  belongs  to  no  animal; 
then  comes  the  other  premiss : 

(1)  animal  belongs  to  all  man. 
(4)  and  (1)  are  the  premisses  of  Celarent,  from  which  follows  the 

(5)  stone  belongs  to  no  man.  This  is  converted,  and  so  the  desired 
conclusion  is  obtained. 

It  is  important  to  notice  that  this  conclusion  is  first  stated  by 
Aristotle  at  the  end  of  the  process  of  proof. 

on  13.10:  Festino:  The  proof  of  this  mood  is  by  reduction  to  Ferio 
(13.06),  the  major  premiss  being  converted  as  in  Cesare  (13.08). 
on  13.11:  Baroco:  If  (1)  Greek  belongs  to  all  Athenian 

and  (2)  to  some  logician  Greek  does  not  belong, 
then  (3)   to  some  logician  Athenian   does  not 
The  proof  proceeds  by  first  hypothesizing  the  conclusion  as  false, 
i.e.  its  contradictory  opposite  is  supposed : 

(4)  Athenian  belongs  to  all  logician. 
Now  comes  the  first  (major)  premiss: 

(5)  Greek  belongs  to  all  Athenian. 
From  these  one  obtains  a  syllogism  in  Barbara  (13.02): 

If  (5)  Greek  belongs  to  all  Athenian 
and  (4)  Athenian  belongs  to  all  logician, 
then  (6)  Greek  belongs  to  all  logician. 
But  the  conclusion  (6)  of  this  syllogism  is  contradictorily  opposed 
to  the  minor  premiss  of  Baroco,  (2),  and  as  this  is  accepted,  the  former 
must  be  rejected.  So  one  of  the  premisses  (4)  and  (5)  must  be  rejected : 
as  (5)  is  accepted,  (4)  must  be  rejected;  and  so  we  obtain  the  con- 
tradictory opposite  of  (4),  namely  (3). 



on  13.12:  Aristotle  defines  the  third  figure,  in  which  the  middle 
term  is  subject  in  both  premisses.  He  considers  the  same  three 
cases  as  in  13.07. 

on  13.13:  Darapti:     If  Greek  belongs  to  all  Athenian 
and  man  belongs  to  all  Athenian, 
then  Greek  belongs  to  some  man. 
This  syllogism  is  first  reduced  to  Darii  (13.05)  by  conversion  of  the 
minor  premiss  -  just  as  Cesare  (13.08)  is  reduced  to  Celarent  (13.03). 
But  Aristotle  then  develops  two  further  methods  of  proof:  a  process 
-  as  with  Baroco  (13.11)  -  'through  the  impossible',  and  the  'setting 
out  of  terms'.  This  last  consists  in  singling  out  a  part,  perhaps  an 
individual  (but  this  is  debated  in  the  literature),  from  the  Athenians, 
say  Socrates.  It  results  that  as  Greek  as  well  as  man  belongs  to  all 
Athenian,  Socrates  must  be  Greek  as  well  as  man.  Therefore  this  is 
a  Greek  who  is  man.  Accordingly  Greek  belongs  to  (at  least)  one 

on  3.14:  Felapton:     If  Egyptian  belongs  to  no  Athenian 
and  man  belongs  to  all  Athenian, 
then  to  some  man  Egyptian  does  not  belong. 
The  proof  proceeds  by  conversion  of  the  minor  (second)  premiss, 
resulting  in  Ferio  (13.06). 

on  3.15:  Disamis:  If  Athenian  belongs  to  all  Greek 

and  logician  belongs  to  some  Greek, 
then  logician  belongs  to  some  Athenian. 
The  first  thing  to  be  noticed  is  that  Aristotle  here  writes  the  minor 
premiss  first,  as  also  in  13.16  and  13.17.  The  proof  is  by  reduction 
to  Darii  (13.05),  just  as  Camestres  (13.09)  was  reduced  to  Celarent 

on  13.16:  Datisi:    If  logician  belongs  to  some  Greek 
and  Athenian  belongs  to  all  Greek, 
then    Athenian     belongs    to     some     logician. 
The  proof  proceeds  by  reduction  to  Darii  (13.05),  the  minor  premiss 
(here  the  first!)  being  converted. 

on  13.17:  Bocardo:     If  Greek  belongs  to  all  Athenian 

and    to    some    Athenian    logician    does    not 

then  to  some  Greek  logician  does  not  belong. 
The  premisses  are  again  in  reversed  order,  the  minor  coming  first. 
The  proof  proceeds  by  reduction  to  the  impossible,  with  use  of 
Barbara  (13.02)  as  in  the  case  of  Baroco  (13.11).  A  proof  by  setting 
out  of  terms  is  also  recommended,  but  not  carried  through, 
on  13.18:  Ferio:  If  Egyptian  belongs  to  no  Greek 

and  logician  belongs  to  some  Greek, 
then  to  some  logician  Egyptian  does  not  belong. 
Aristotle  reduces  this  syllogism  to  Ferio  (13.06),  by  conversion  of 
the  minor  (second)  premiss. 



This  paraphrase  with  comments  is,  be  it  noted,  a  concession  to  the 
modern  reader.  For  in  his  Analytics  Aristotle  never  argued  by  means 
of  substitutions,  as  we  have  been  doing,  except  in  proofs  of  invalidity. 
However,  in  view  of  the  contemporary  state  of  logical  awareness,  it 
seems  necessary  to  elucidate  the  text  in  this  more  elementary  way. 


If  we  consider  the  passages  on  which  we  have  just  commented, 
the  first  thing  we  notice  is  that  the  definition  which  Aristotle  gives  of 
the  syllogism  (10.05),  does  indeed  contain  it,  but  is  much  too  wide: 
the  analytic  syllogism  as  we  call  the  class  of  formulae  considered  in 
chapters  4-6  of  the  first  book  of  the  Prior  Analytics,  can  be  described 
as  follows: 

1.  It  is  a  conditional  sentence,  the  antecedent  of  which  is  a  con- 
junction of  two  premisses.  Its  general  form  is :  '  If  p  and  q,  then  r',  in 
which  propositional  forms  are  to  be  substituted  for  'p',  'g'  and  V.  So 
the  Aristotelian  syllogism  has  not  the  later  form:  'p;  q;  therefore  r', 
which  is  a  rule.  The  Aristotelian  syllogism  is  not  a  rule  but  a  proposition. 

2.  The  three  propositional  forms  whose  inter-connection  produces  a 
syllogism,  are  always  of  one  of  these  four  kinds:  'A  belongs  to  all  B\ 

A  belongs  to  no  B\  lA  belongs  to  (at  least)  some  B\  '  (at  least)  to 
some  B  A  does  not  belong'.  Instead  of  this  last  formula,  there  some- 
times occurs  the  (equivalent)  one:  'A  belongs  not  to  all  B\  The 
word  'necessary'  or  'must'  is  often  used :  evidently  that  only  means 
here  that  the  conclusion  in  question  follows  logically  from  the 

3.  Where  we  have  been  speaking  not  of  propositions  but  of 
forms,  seeing  that  Aristotelian  syllogisms  always  contain  letters 
('A',  'B',  T'  etc.)  in  place  of  words,  which  are  evidently  to  be 
interpreted  as  variables,  Aristotle  himself  gives  examples  of  how 
substitutions  can  be  made  in  them.  That  is  indeed  the  only  kind 
of  substitution  for  variables  known  to  him:  he  has,  for  example, 
no  thought  of  substituting  variables  for  variables.  Nevertheless 
this  is  an  immense  discovery :  the  use  of  letters  instead  of  constant 
words  gave  birth  to  formal  logic. 

4.  In  every  syllogism  we  find  six  such  letters,  called  'terms' 
(opot.,  'boundaries'),  equiform  in  pairs.  Aristotle  uses  the  following 
terminology:  the  term  which  is  predicate  in  the  conclusion  and  the 
term  equiform  to  it  in  one  of  the  premisses,  are  called  'major', 
evidently  because  in  the  first  figure  -  but  there  only  -  this  has  the 
greatest  extension.  The  term  which  is  subject  in  the  conclusion,  and 
the  term  equiform  to  it  in  one  of  the  premisses,  he  calls  a  'minor'  or 
'last'  term  (eXocttov,  Ict^octov)  for  the  same  reason.  Finally  the  two 
equiform  terms  that  occur  only  in  the  premisses  are  called  'middle' 
terms.   By  contrast  the   two  others   are   called    'extremes'   (axpa). 



Admittedly  the  terms  are  not  so  defined  in  the  text  of  Aristotle :  he 
gives  complicated  definitions  based  on  the  meaning  of  the  terms ;  but 
his  customary  syllogistic  practice  keeps  to  the  foregoing  definitions. 
-  Sometimes  the  premiss  containing  the  major  term  is  called  'the 
major',  the  other  'the  minor'. 

5.  The  letters  (variables)  can,  in  the  system,  only  be  substituted  by 
universal  terms;  they  are  term-variables  for  universal  terms.  But  it 
would  not  be  right  to  call  them  class-variables,  for  that  would  be  to 
ascribe  to  Aristotle  a  distinction  between  intension  and  extension 
which  is  out  of  place.  One  may  ask  why  the  founder  of  logic,  whose 
philosophical  development  proceeded  steadily  away  from  Platonism 
towards  a  recognition  of  the  importance  of  the  individual,  completely 
omitted  singular  terms  in  what  (by  contrast  to  the  Hermeneia)  is 
his  most  mature  work.  The  reason  probably  lies  in  his  assumption 
that  such  terms  are  not  suitable  as  predicates  (13.19),  whereas 
syllogistic  technique  requires  every  extreme  term  to  occur  at  least 
once  as  predicate. 

6.  It  is  usually  said  that  a  further  limitation  is  required,  namely 
that  void  terms  must  not  be  substituted  for  the  variables.  But  this  is 
only  true  in  the  context  of  certain  interpretations  of  the  syllogistic; 
on  other  interpretations  this  limitation  is  not  required. 

To  sum  up :  we  have  in  the  syllogistic  a  formal  system  of  term-logic, 
with  variables,  limited  to  universal  terms,  and  consisting  of  propo- 
sitions, not  rules. 

This  system  is  also  axiomatized.  Hence  we  have  here  together 
three  of  the  greatest  discoveries  of  our  science:  formal  treatment, 
variables,  and  axiomatization.  What  makes  this  last  achievement 
the  more  remarkable  is  the  fact  that  the  system  almost  achieves 
completeness  (there  is  lacking  only  an  exact  elaboration  of  the 
moods  of  the  fourth  figure).  This  is  something  rare  for  the  first 
formulation  of  a  quite  original  logical  discovery. 


The  syllogisms  are  divided  into  three  classes  (ax^aTa),  'figures'  as 
this  was  later  translated,  according  to  the  position  of  the  middle  term. 
According  to  Aristotle  there  are  only  three  such  figures : 

13.20  So  we  must  take  something  midway  between  the 
two,  which  will  connect  the  predications,  if  we  are  to  have  a 
syllogism  relating  this  to  that.  If  then  we  must  take  something 
common  in  relation  to  both,  and  this  is  possible  in  three  ways 
(either  by  predicating  A  of  C,  and  C  of  B,  or  C  of  both,  or 
both  of  C),  and  these  are  the  figures  of  which  we  have  spoken, 
it  is  clear  that  every  syllogism  must  be  made  in  one  or  other  of 
these  figures. 



But  evidently  this  is  so  far  from  being  the  case  that  Aristotle 
himself  was  well  aware  of  a  fourth  figure.  He  treats  its  syllogisms 
as  arguments  obtainable  from  those  already  gained  in  the  first 

13.21  It  is  evident  also  that  in  all  the  figures,  whenever  a 
proper  syllogism  does  not  result,  if  both  the  terms  are  affir- 
mative or  negative  nothing  necessary  follows  at  all,  but  if  one 
is  affirmative,  the  other  negative,  and  if  the  negative  is  stated 
universally,  a  syllogism  always  results  relating  the  minor  to 
the  major  term,  e.g.  if  A  belongs  to  all  or  some  B,  and  B 
belongs  to  no  C:  for  if  the  premisses  are  converted  it  is 
necessary  that  C  does  not  belong  to  some  A.  Similarly  also  in 
the  other  figures. 

The  case  under  consideration  is  this:  the  premisses  are  (1)  lA 
belongs  to  all  B\  (2)  'B  belongs  to  no  C\  They  are  both  converted  and 
their  order  is  reversed  (an  operation  that  is  irrelevant  for  Aristotle), 
so  that  we  get:  lC  belongs  to  no  B'  and  'B  belongs  to  some  A\  But 
those  are  the  premisses  of  the  fourth  mood  of  the  first  figure  (Ferio, 
13.06),  which  has  as  conclusion  'to  some  A  C  does  not  belong'.  Now 
if  'major  term'  and  'minor  term'  are  defined  as  has  been  done  above 
(§  13,  C  4)  in  accordance  with  the  practice  of  Albalag  and  the  moderns, 
then  evidently  C  is  the  major  term,  A  the  minor,  and  so  (2)  is  the 
major  premiss,  (1)  the  minor  premiss.  From  that  it  follows  that  the 
middle  term  is  predicate  in  the  major  premiss,  subject  in  the  minor, 
just  the  reverse  of  the  situation  in  the  first  figure.  We  have  here 
therefore  a  fourth  figure.  That  Aristotle  refuses  to  recognize  any 
such,  is  due  to  his  not  giving  a  theoretical  definition  of  the  terms 
according  to  their  place  in  the  conclusion,  but  according  to  their 
extension,  and  so  not  a  formal  definition  but  one  dependent  on  their 
meaning.  -  The  syllogism  just  investigated  later  came  to  be  called 

Aristotle  explicitly  stated  two  syllogism  of  this  figure,  Fresison 
already  cited,  and  in  the  same  passage  (13.31)  Fesapo;  he  hints  only 
at  three  more  (13.32):  Dimaris,  Bamalip  and  Camenes. 

The  same  text  (13.22)  would  permit  us  to  gain  still  other  syllo- 
gisms from  two  of  the  second  figure  Cesare:  13.08  and  Cameslres: 
13.09)  and  from  three  of  the  third  {Darapti:  13.13,  Disamis:  13.15 
and  Dalisi:  13.16).  It  is  worth  remarking  that  these  hints  do  not 
occur  in  the  text  of  the  proper  presentation  of  the  syllogistic; 
apparently  Aristotle  only  made  these  discoveries  when  his  system 
was  already  in  being. 

Consequently  the  following  passage  seems  to  be  a  later  addition. 
It  is  at  the  origin  of  what  later  came  to  be  called  the  'subalternate 
syllogisms',  Barbari,  Celaront,  Cesaro,  Camestrop  and  Calemop. 



13.23  It  is  possible  to  give  another  reason  concerning 
those  (syllogisms)  which  are  universal.  For  all  the  things  that 
are  subordinate  to  the  middle  term  or  to  the  conclusion  may 
be  proved  by  the  same  syllogism,  if  the  former  are  placed  in 
the  middle,  the  latter  in  the  conclusion;  e.g.  if  the  conclusion 
AB  is  proved  through  C,  whatever  is  subordinate  to  B  or  C 
must  accept  the  predicate  A :  for  if  D  is  included  in  B  as  in  a 
whole,  and  B  is  included  in  A,  then  D  will  be  included  in  A. 
Again  if  E  is  included  in  C  as  in  a  whole,  and  C  is  included  in 
A,  then  E  will  be  included  in  A.  Similarly  if  the  syllogism  is 
negative.  In  the  second  figure  it  will  be  possible  to  infer  only 
that  which  is  subordinate  to  the  conclusion,  e.g.  if  A  belongs 
to  no  B  and  to  all  C ;  we  conclude  that  B  belongs  to  no  C.  If 
then  D  is  subordinate  to  C,  clearly  B  does  not  belong  to 
it.  But  that  B  does  not  belong  to  what  is  subordinate  to  A,  is 
not  clear  by  means  of  the  syllogism. 

These  syllogisms,  however,  are  not  developed. 

If  we  want  to  summarize  the  content  of  the  texts  we  have  adduced, 
we  see  that  Aristotle  in  fact  expressly  formulated  the  conditions 
required  for  a  system  of  twenty-four  syllogistic  moods,  six  in  each 
figure.  Of  these  twenty-four  he  only  developed  nineteen  himself, 
only  fourteen  of  them  thoroughly.  The  remaining  ten  fall  into  three 
classes:  (1)  exactly  formulated  (Fesapo,  Fresison:  13.21):  (2)  not 
formulated,  but  clearly  indicated  (Dimaris,  Bamalip,  Camenes: 
13.22);  (3)  only  indirectly  indicated:  the  five  'subalternate'  moods 

That  explains  how  historically  sometimes  fourteen,  sometimes 
nineteen  and  again  at  other  times  twenty-four  moods  are  spoken  of. 
The  last  figure  is  obviously  the  only  correct  one.  For  evidently  no 
systematic  principle  can  be  derived  from  the  fact  that  the  author  of 
the  syllogistic  did  not  precisely  develop  certain  moods. 



Aristotle  axiomatized  the  syllogistic,  and  in  more  than  one  way.  In 
this  connection  we  shall  first  cite  the  most  important  passages  in 
which  he  presents  his  theory  of  the  system  as  axiomatized,  and  then 
give  the  axiomatization  itself.  For  this  theory  is  the  first  of  its  kind 
known  to  us,  and  notwithstanding  its  weaknesses,  must  be  consi- 
dered as  yet  another  quite  original  contribution  made  to  logic  by 



Aristotle.  The  point  is  a  methodological  one  of  course,  not  a  matter  of 

formal  logic,  and  that  Aristotle  was  himself  aware  of: 

14.01  We  now  state  by  what  means,  when,  and  how  every 
syllogism  is  produced;  subsequently  we  must  speak  of  demon- 
stration. Syllogism  should  be  discussed  before  demonstration, 
because  syllogism  is  the  more  general:  demonstration  is  a 
sort  of  syllogism,  but  not  every  syllogism  is  a  demonstration. 

Aristotle's  doctrine  of  demonstration  is  precisely  his  methodology. 
But  as  the  methodology  of  deduction  is  closely  connected  with  formal 
logic,  we  must  go  into  at  least  a  few  details. 

14.02  We  suppose  ourselves  to  possess  unqualified  scientific 
knowledge  of  a  thing,  as  opposed  to  knowing  it  in  the  accidental 
way  in  which  the  sophist  knows,  when  we  think  that  we  know 
the  cause  on  which  the  fact  depends,  as  the  cause  of  that  fact 
and  of  no  other,  and,  further,  that  the  fact  could  not  be  other 
than  it  is  .  .  .  There  may  be  another  manner  of  knowing  as 
well  -  that  will  be  discussed  later.  What  I  now  assert  is  that 
at  all  events  we  do  know  by  demonstration.  By  demonstration 
I  mean  a  syllogism  productive  of  scientific  knowledge,  a 
syllogism,  that  is,  the  grasp  of  which  is  eo  ipso  such  knowledge. 
Assuming  then  that  my  thesis  as  to  the  nature  of  scientific 
knowing  is  correct,  the  premisses  of  demonstrated  knowledge 
must  be  true,  primary,  immediate,  better  known  than  and 
prior  to  the  conclusion,  which  is  further  related  to  them  as 
effect  to  cause.  Unless  these  conditions  are  satisfied,  the  basic 
truths  will  not  be  'appropriate'  to  the  conclusion.  Syllogism 
there  may  indeed  be  without  these  conditions,  but  such 
syllogism,  not  being  productive  of  scientific  knowledge,  will 
not  be  demonstration. 

14.03  There  are  three  elements  in  demonstration  :  (1) 
what  is  proved,  the  conclusion  -  an  attribute  inhering  essen- 
tially in  a  genus;  (2)  the  axioms,  i.e.  the  starting  points  of 
proof;  (3)  the  subject-genus  whose  attributes,  i.e.  essential 
properties,  are  revealed  by  the  demonstration. 

It  emerges  clearly  from  this  text  that  for  Aristotle  a  demonstration 
(1)  is  a  syllogism,  (2)  with  specially  constructed  premisses,  and  (3) 
with  a  conclusion  in  which  a  property  (11.081  is  predicated  of  a  genus. 
That,  however,  can  only  be  achieved  by  means  of  a  syllogism  in  the 
first  figure : 



14.04  Of  all  figures  the  most  scientific  is  the  first.  Thus,  it 
is  the  vehicle  of  the  demonstrations  of  all  the  mathematical 
sciences,  such  as  arithmetic,  geometry,  and  optics,  and  practi- 
cally of  all  sciences  that  investigate  causes  ...  a  second  proof 
that  this  figure  is  the  most  scientific;  for  grasp  of  a  reasoned 
conclusion  is  the  primary  condition  of  knowledge.  Thirdly, 
the  first  is  the  only  figure  which  enables  us  to  pursue  know- 
ledge of  the  essence  of  a  thing.  .  .  .  Finally,  the  first  figure  has 
no  need  of  the  others,  while  it  is  by  means  of  the  first  that  the 
other  two  figures  are  developed,  and  have  their  intervals 
close-packed  until  immediate  premisses  are  reached.  Clearly, 
therefore,  the  first  figure  is  the  primary  condition  of  knowledge. 

This  doctrine  is  only  of  historical  importance,  though  that  is 
considerable:  on  the  other  hand  the  essential  of  Aristotle's  views  on 
the  structure  of  an  axiomatic  system  has  remained  a  part  of  every 
methodology  of  deduction  right  to  our  own  day: 

14.05  Our  own  doctrine  is  that  not  all  knowledge  is  demon- 
strative: on  the  contrary,  knowledge  of  the  immediate 
premisses  is  independent  of  demonstration.  (The  necessity  of 
this  is  obvious;  for  since  we  must  know  the  prior  premisses 
from  which  the  demonstration  is  drawn,  and  since  the  regress 
must  end  in  immediate  truths,  those  truths  must  be  indemon- 
strable.) .  .  Now  demonstration  must  be  based  on  premisses 
prior  to  and  better  known  than  the  conclusion ;  and  the  same 
things  cannot  simultaneously  be  both  prior  and  posterior  to 
one  another:  so  circular  demonstration  is  clearly  not  possible 
in  the  unqualified  sense  of  'demonstration',  but  only  possible 
if  'demonstration'  be  extended  to  include  that  other  method 
of  argument  which  rests  on  a  distinction  between  truths 
prior  to  us  and  truths  without  qualification  prior,  i.e.  the 
method  by  which  induction  produces  knowledge.  .  .  .  The 
advocates  of  circular  demonstration  are  not  only  faced  with 
the  difficulty  we  have  just  stated:  in  addition  their  theory 
reduces  to  the  mere  statement  that  if  a  thing  exists,  then  it 
does  exist  -  an  easy  way  of  proving  anything.  That  this  is 
so  can  be  clearly  shown  by  taking  three  terms,  for  to  constitute 
the  circle  it  makes  no  difference  whether  many  terms  or  few 
or  even  only  two  are  taken.  Thus  by  direct  proof,  if  A  is,  B 
must  be ;  if  B  is,  C  must  be ;  therefore  if  A  is,  C  must  be.  Since 
then  -  by  the  circular  proof  -  if  A  is,  B  must  be,  and  if  B  is,  A 
must  be,  A  may  be  substituted  for  C  above.  Then  'if  B  is,  A 



must  be'  =  'if  B  is,  C  must  be',  which  above  gave  the  con- 
clusion 'if  A  is,  C  must  be',  but  C  and  A  have  been  identified. 

This  is,  be  it  said  at  once,  far  the  clearest  passage  about  our 
problem,  which  evidently  faced  Aristotle  with  enormous  difficulties. 
Two  elements  are  to  be  distinguished :  on  the  one  hand  it  is  a  matter 
of  epislemological  doctrine,  according  to  which  all  scientific  knowledge 
must  finally  be  reduced  to  evident  and  necessary  premisses;  on  the 
other  hand  is  a  logical  theory  of  deduction,  which  states  that  one 
cannot  demonstrate  all  sentences  in  a  system,  but  must  leave  off 
somewhere;  for  neither  a  processus  in  infinitum  nor  a  circular 
demonstration  is  possible.  In  other  words:  there  mu-t  be  axioms  in 
every  system. 


This  doctrine  was  now  applied  by  Aristotle  to  formal  logic  itself, 
i.e.  to  the  syllogistic ;  yes,  the  syllogistic  is  the  first  known  axiomatic 
system,  or  more  precisely  the  first  class  of  such  systems:  for  Ari- 
stotle axiomatized  it  in  several  ways.  One  can  distinguish  in  his 
work  the  following  systems:  1)  with  the  four  syllogisms  of  the  first 
figure  (together  with  other  laws)  as  axioms,  2)  with  the  first  two 
syllogisms  of  the  same  figure,  3)  with  syllogisms  of  any  figure  as 
axioms,  in  which  among  other  features  the  syllogisms  of  the  first 
figure  are  reduced  to  those  of  the  second  and  third.  These  three 
systems  are  presented  in  an  object-language;  there  is  further  to  be 
found  in  Aristotle  a  sketch  for  the  axiomatization  of  the  syllogistic 
in  a  metalanguage. 

We  take  first  the  second  system,  the  first  having  been  fully 
presented  above  in  §  13. 

14.06  It  is  possible  also  to  reduce  all  syllogisms  to  the 
universal  syllogisms  in  the  first  figure.  Those  in  the  second 
figure  are  clearly  made  perfect  by  these,  though  not  all  in  the 
same  way;  the  universal  syllogisms  are  made  perfect  by  con- 
verting the  negative  premiss,  each  of  the  particular  syllogisms 
by  reductio  ad  impossibile.  In  the  first  figure  particular 
syllogisms  are  indeed  made  perfect  by  themselves,  but  it  is 
possible  also  to  prove  them  by  means  of  the  second  figure, 
reducing  them  ad  impossibile,  e.g.  if  A  belongs  to  all  B,  and 
B  to  some  C,  it  follows  that  A  belongs  to  some  C.  For  if  it 
belonged  to  no  67,  and  belongs  to  all  B,  then  B  will  belong 
to  no  C:  this  we  know  by  means  of  the  second  figure. 

It  is  here  shewn  that  Darii  (13.05)  can  be  reduced  to  Cameslres 
(13.09) ;  the  proof  of  Ferio  (13.06)  is  similarly  effected,  and  it  is  then 



shewn  that  the  syllogisms  of  the  third  figure  can  also  be  easily 

In  these  operations  the  syllogisms  of  the  first  figure  always  play 
the  part  of  axioms,  for  the  reason  that  they  are  to  be  'perfect' 
(TeAeLOi)  syllogisms  (14.08).  This  expression  is  explained  thus: 

14.09  I  call  that  a  perfect  syllogism  which  needs  nothing 
other  than  what  has  been  stated  to  make  plain  what  neces- 
sarily follows;  a  syllogism  is  imperfect,  if  it  needs  either  one 
or  more  propositions,  which  are  indeed  the  necessary  conse- 
quences of  the  terms  set  down,  but  have  not  been  expressly 
stated  as  premisses. 

But  this  can  only  mean  that  perfect  syllogisms  are  intuitively 


To  be  able  to  deduce  his  syllogisms  Aristotle  makes  use  of  three 
procedures,  and  in  each  of  another  class  of  formulae,  not  named  as 
axioms  and  in  part  tacitly  presented.  The  procedures  are  direct 
proof  (SzixTitic,  avaysLv),  reduction  to  the  impossible  (zlc,  to  aSuvarov 
avayetv)  and  ecthesis  (setting  out  of  terms,  exOsctic;). 

In  direct  proof  the  laws  of  conversion  of  sentences  are  explicitly 
presupposed ;  they  are  three : 

14.10  If  A  belongs  to  no  B,  neither  will  B  belong  to  any  A. 
For  if  to  some,  say  to  C,  it  will  not  be  true  that  A  belongs  to 
no  B;  for  C  is  one  of  the  things  (which  are)  B. 

14.11  If  A  belongs  to  all  B,  B  also  will  belong  to  some  A; 
for  if  to  none,  then  neither  will  A  belong  to  any  B;  but  by 
hypothesis  it  belonged  to  all. 

14.12  If  A  belongs  to  some  B,  B  also  must  belong  to  some 
A;  for  if  to  none,  then  neither  will  A  belong  to  any  B. 

Aristotle  prefaces  the  syllogistic  proper  with  these  laws  and  their 
justification,  clearly  conscious  that  he  needs  them  for  the  'direct 
procedure'  .They  are  the  laws  of  conversion  of  affirmative  (universal 
and  particular),  and  universal  negative  propositions.  (The  conversion 
of  particular  negatives  is  expressly  recognized  as  invalid:  14.13.) 

It  is  noteworthy  that  Aristotle  tries  to  axiomatize  these  laws  too : 
the  first  is  proved  by  ecthesis  and  serves  as  axiom  for  the  two 

Besides  these  explicit  presuppositions  of  the  syllogistic,  some 
rules  of  inference  are  also  used,  without  Aristotle  having  consciously 
reflected  on  them.  They  are  these : 

14.141  Should  'If  p  and  q,  then  r'  and  'If  s,  then  p'  be 
valid,  then  also  'If  s  and  q,  then  r'  is  valid. 



14.142  Should  'If  p  and  q,  then  r'  and  'If  s,  then  qf  be 
valid,  then  also  'If  p  and  s,  then  r'  is  valid. 

14.151  Should  'If  p  and  q,  then  r'  be  valid,  then  also  'If  q 
and  p,  then  r'  is  valid. 

14.161  Should  'If  p,  then  </'  and  'If  </,  then  r'  be  valid,  then 
also  'If  p,  then  r'  is  valid. 

Some  of  these  rules  also  used  -  without  being  explicitly  appealed 
to  -  for  constructing  formulae  that  later  came  to  be  called  'polysyllo- 
gisms,  or  'soriteses' : 

14.17  It  is  clear  too  that  every  demonstration  will  proceed 
through  three  terms  and  no  more,  unless  the  same  con- 
clusion is  established  by  different  pairs  of  propositions;  .  .  . 
Or  again  when  each  of  the  propositions  A  and  B  is  obtained 
by  syllogistic  inference,  e.g.  A  by  means  of  D  and  E,  and 
again  B  by  means  of  F  and  G.  .  .  .  But  thus  also  the  syllogisms 
are  many;  for  the  conclusions  are  many,  e.g.  A  and  B  and  C. 

Also  to  be  noted  in  this  text  is  Aristotle's  evident  use  of  proposi- 
tional  variables. 


Aristotle  has  two  different  procedures  for  reduction  to  the  impos- 
sible, the  first  being  invalid  and  clearly  earlier.  In  both,  the  laws  of 
opposition  are  presupposed.  By  contrast  to  the  laws  of  conversion 
these  are  neither  systematically  introduced  nor  axiomatized;  they 
occur  as  the  occassion  of  the  deduction  requires.  The  reason  for 
their  not  being  systematized  or  axiomatized  may  be  that  the  essential 
points  about  them  have  been  stated  already  in  the  Hermeneia.  The 
main  features  have  been  summarized  above  (12.10,  12,11). 

The  procedures  are  as  follows: 

First  procedure 

It  is  used  to  reduce  Baroco  (13.11)  and  Bocardo  (13.17),  and  takes 
the  course  outlined  above  (in  the  commentary  on  13.11).  As  Luka- 
siewicz (14.18)  has  shown,  it  is  not  conclusive.  This  can  be  made 
evident  by  the  following  substitution:  we  put  'bird'  for  'M',  'beast' 
for  W  and  'owl'  for  lX'  in  Baroco  (13.11).  We  obtain: 
If  (1)  bird  belongs  to  all  beast 
and  (2)  to  some  owl  bird  does  not  belong, 
then  (3)  to  some  owl  beast  does  not  belong. 
The  syllogism  is  correct,  being  a  substitution  in  Baroco;  but  all  its 
three  component  sentences  are  manifestly  false.  Now  if  we  apply  the 
procedure  described  above  (commentary  on  13.11),  we  must  form 
the  contradictory  opposite  to  (3) : 
(4)  Beast  belongs  to  all  owl. 



This  produces  with  (1)  a  syllogism  in  Barbara  (13.02),  having  as 

(5)  Bird  belongs  to  all  owl,  which  so  far  from  being  false,  is 
evidently  true.  Hence  the  procedure  fails  to  give  the  required 
conclusion  and  must  be  deemed  incorrect. 

It  would  certainly  be  correct  if  Aristotle  had  not  expressed  the 
syllogism  as  a  conditional  sentence  (in  which  the  antecedent  does  not 
need  to  be  asserted),  but  in  the  scholastic  manner  as  a  rule  (31.11) 
in  which  one  starts  from  asserted  premisses. 

Second  procedure 

We  do  not  know  whether  Aristotle  saw  the  incorrectness  of  the 
first  procedure;  in  any  case  in  book  B  of  the  Prior  Analytics  he 
several  times  uses  another  which  is  logically  correct. 

It  is  to  be  found  in  the  place  where  he  treats  of  the  so-called 
'conversion'  (avTLaTpocp-y))  of  syllogisms,  a  matter  of  replacing  one 
premiss  by  the  (contradictory)  opposite  of  the  conclusion. 

14.19  Suppose  that  A  belongs  to  no  B,  and  to  some  C: 
the  conclusion  is  BC.  ...  If  the  conclusion  is  converted  into 
its  contradictory,  both  premisses  can  be  refuted.  For  if  B 
belongs  to  all  C,  and  A  to  no  B,  A  will  belong  to  no  C:  but 
it  was  assumed  to  belong  to  some  C. 

The  following  scheme  reproduces  the  thought: 

Original  syllogism  Converted  syllogism 

(Festino)  (Celarent) 

If  A  belongs  to  no  B  If  A  belongs  to  no  B 

and  A  belongs  to  some  C,  and  B  belongs  to  all  C, 

then  to  some  C  B  does  not  then  A  belongs  to  no  C 


The  rule  presupposed,  of  which  Aristotle  was  conscious  (cf. 
16.33)  -  he  often  used  it,  -  is  this: 

14.201  Should  'If  p  and  q,  then  r'  be  valid  then  also  'If  not-r 
and  q,  then  not-p'  is  valid. 

A  similar  rule,  also  often  used,  is: 

14.202  Should  'If  p  and  q,  then  r'  be  valid,  then  also  'If 
p  and  not-r,  then  not-qr'  is  valid. 

By  the  use  of  these  rules  with  the  laws  of  opposition  and  some  of 
the  rules  given  above  (14.151,  14.161),  any  syllogism  can  in  fact  be 
reduced  to  another. 




By  varying  this  second  procedure  Aristotle  was  able  to  construct 
a  third  axiomatization  -  or  rather  a  class  of  further  axiomatiza- 
tions  -  of  his  syllogistic:  Syllogisms  of  either  the  first,  second  or 
third  figure  are  taken,  and  the  others  proved  from  them  by  reduc- 
tion to  the  impossible.  The  result  is  summarized  thus: 

14.21  It  is  clear  that  in  the  first  figure  the  syllogisms  are 
formed  through  the  middle  and  the  last  figures  ...  in  the 
second  through  the  first  and  the  last  figures  ...  in  the  third 
through  the  first  and  the  middle  figures. 

We  shall  not  go  into  the  practical  details  (14.22),  but  only  note 
that  Aristotle  replaces  premisses  not  only  by  their  contradictory, 
but  also  by  their  contrary  opposites,  and  that  he  investigates  all 

The  results  of  replacing  premisses  by  the  contradictory  opposite 
of  the  conclusion  can  be  clearly  presented  in  the  following  way: 
From  a  syllogism  of  figure  1  2         3 

there  results  by  substitution 
of  the  negation  of  the  conclusion 
a  syllogism  of  figure: 

substitution  for  the  major  premiss  3         3         1 

substitution  for  the  minor  premiss  2         12 


A  word  must  now  be  said  about  the  'dictum  de  omni  el  nullo'  that 
later  became  so  famous.  It  concerns  the  following  sentence: 

14.23  That  one  thing  should  be  in  the  whole  of  another  and 
should  be  predicated  of  all  of  another  is  the  same.  We  say 
that  there  is  predication  of  all  when  it  is  impossible  to  take 
anything  of  which  the  other  will  not  be  predicated;  and 
similarly  predication  of  none. 

It  is  not  clear  whether  Aristotle  really  intended  here  to  establish 
an  axiom  for  his  system,  as  has  often  been  supposed.  One  is  rather 
led  to  suppose  that  he  is  simply  describing  the  first  and  second 
moods  of  the  first  figure  (13.02,  13.03).  However,  the  dictum  can 
be  understood  as  an  axiom  if  it  is  considered  as  a  summary  of  the 
first  four  moods  of  the  first  figure,  which  is  not  in  itself  impossible. 

In  this  connection  we  quote  a  historically  and  systematically 
more  important  passage  in  which  Aristotle  deals  with  a  problem  of 
the  theory  of  the  three  figures  (13.20).  In  it  he  makes  an  essential 
advance  in  analyzing  propositions  and  gives  expression  to  thoughts 
that  are  not  without  significance  for  the  theory  of  quantification. 



14.24  It  is  not  the  same,  either  in  fact  or  in  speech,  that  A 
belongs  to  all  of  that  to  which  B  belongs,  and  that  A  belongs 
to  all  of  that  to  all  of  which  B  belongs :  for  nothing  prevents  B 
from  belonging  to  C,  though  not  to  all  C:  e.g.  let  B  stand 
for  'beautiful'  and  C  for  'white'.  If  beauty  belongs  to  something 
white,  it  is  true  to  say  that  beauty  belongs  to  that  which  is 
white:  but  not  perhaps  to  everything  that  is  white. 

Here  an  analysis  of  the  sentence  A  belongs  to  all  B  'is  presented, 
which  could  be  interpreted  in  this  way:  'For  all  x:  if  B  belongs  to  x, 
then  A  belongs  to  x;  it  would  then  be  a  matter  of  the  modern  formal 
implication.  That  Aristotle  thought  of  such  an  analysis  -  at  least 
during  his  later  period  -  seems  guaranteed  by  the  fact  that  he 
explicitly  applied  it  to  modal  logic  (cf.  15.13).  The  Scholastics,  as  we 
shall  see,  treated  these  thoughts  as  an  elucidation  of  the  dictum. 


Aristotle  also  described  his  syllogisms  metalogically  in  such  a  way 
that  a  new,  metalogical  system  could  easily  be  established: 

14.25  In  every  syllogism  one  of  the  premisses  must  be 
affirmative,  an  universality  must  be  present. 

14.26  It  is  clear  that  every  demonstration  will  proceed 
through  three  terms  and  no  more,  unless  the  same  conclusion 
is  established  by  different  pairs  of  propositions. 

14.27  This  being  evident,  it  is  clear  that  a  syllogistic 
conclusion  follows  from  two  premisses  and  not  from  more 
than  two. 

14.28  And  it  is  clear  also  that  in  every  syllogism  either 
both  or  one  of  the  premisses  must  be  like  the  conclusion.  I 
mean  not  only  in  being  affirmative  or  negative,  but  also  in 
being  necessary,  assertoric  or  contingent. 

Aristotle  does  not  carry  out  this  application  to  modal  logic; 
possibly  this  is  an  interpolation  by  another  hand. 

In  developing  the  several  figures  Aristotle  established  similar 
rules  for  each.  13.05  contains  an  example.  Taken  all  together  these 
rules  form  an  almost  complete  metalogical  description  of  the  syllo- 
gistic, which  one  would  like  to  develop. 


Here  we  want  to  allude  briefly  to  a  doctrine  of  the  Prior  Analytics 
which  is  not  essentially  a  matter  of  formal  logic  but  rather  of 
methodology,   and   that  is  the   discussions   about  what  was   later 



called  the  inventio  medii.   In  connection  with  axiomatization  one 

can  ask  two  different  b;isic  questions:  (1)  What  follows  given 
premisses?  (2)  From  what  premisses  can  ;i  given  sentence  (con- 
clusion) be  deduced?  Aristotle  primarily  considered  the  first  ques- 
tion, but  in  the  following  text  and  its  continuation  he  poses  also 
the  second,  and  tries  to  show  the  premisses  of  a  syllogism  must  be 
constructed  in  order  to  yield  a  given  conclusion.  At  the  same  time 
he  gives  practical  advice  on  the  forming  of  syllogisms: 

14.29  The  manner  in  which  every  syllogism  is  produced, 
the  number  of  the  terms  and  premisses  through  which  il 
proceeds,  the  relation  of  the  premisses  to  one  another,  the 
character  of  the  problem  proved  in  each  figure,  and  the  num- 
ber of  the  figures  appropriate  to  each  problem,  all  these 
matters  are  clear  from  what  has  been  said.  We  must  now 
state  how  we  may  ourselves  always  have  a  supply  of  syllogisms 
in  reference  to  the  problem  proposed  and  by  what  road  we  may 
reach  the  principles  relative  to  the  problem:  for  perhaps  we 
ought  not  only  to  investigate  the  construction  of  syllogisms, 
but  also  to  have  the  power  of  making  them. 

We  do  not  need  to  pursue  the  details  of  this  theory  here.  It  only 
interests  us  as  the  starting-point  of  the  scholastic  pons  asinorum. 

§  15.  MODAL  LOGIC* 


Aristotle  distinguishes  three  principal  classes  of  premisses: 

15.01  Every  premiss  is  either  about  belonging  to,  or 
necessarily  belonging  to,  or  possibly  belonging  to. 

The  expressions  'necessary'  (el*  avayxY)*;)  and  'possible'  (can 
belong  to,  IvSe^ETai,  SuvaTai)  have  several  meanings. 

1.  In  respect  of  the  functor  'necessary'  (or  'must')  we  have  already 
remarked  (§  13,  C,  2)  that  it  often  only  expresses  logical  consequence. 

*  The  Aristotelian  (as  also  the  Theophrastan)  modal  logic  is  here  interpreted  in 
the  way  that  was  customary  among-  the  Scholastics,  rediscovered  by  A.  Becker 
in  1934,  and  served  as  a  basis  for  my  ideas  in  works  on  the  history  of  modal 
logic,  on  Theophrastus  and  on  ancient  logic. 

However,  while  I  was  writing  this  work,  Prof.  J.  Lukasiewicz  communicated 
his  being  in  possession  of  a  quite  different  interpretation,  showing  the  Aristote- 
lian system  to  have  contained  mistakes  which  were  rectified  by  Theophrastus. 
This  new  interpretation  has  now  been  published  [Aristotle's  Syllogistic,  2nd  ed., 
Oxford,  1957). 



That  this  is  so,  can  be  clearly  seen  where  for  instance  Aristotle 
says  'It  is  necessary  that  A  necessarily  belongs  to  B'  (15.02),  or 
again  'It  is  necessary  that  A  possibly  belongs  to  B'  (15.03).  The 
first  'necessary'  evidently  means  logical  (a.n'k&c,)  and  hypothetical 
(toutcov  ovtcov)  necessity  (15.04).  The  necessity  that  something  is 
when  it  is  (6t<xv  JJ)  obviously  belongs  to  the  second  of  these  classes 

2.  Even  the  simple,  unqualified  (assertoric)  'belonging  to',  which 
Aristotle  often  calls  'mere  belonging  to',  is  divided  into  an  absolute 
(a.Tzk&Q)  and  temporally  qualified  (xorra  ^povov)  kind,  with  different 
logical  properties  (15.06). 

3.  As  to  possibility,  Aristotle  distinguishes  at  first  two  kinds: 
the  one-sided  and  the  two-sided. 

This  distinction  emerges  from  a  searching  discussion  which 
Aristotle  conducts  in  the  Hermeneia.  The  passage  is  of  great  impor- 
tance for  the  understanding  of  the  whole  doctrine  of  modalities, 
and  so  we  give  it  in  full : 

15.07  When  it  is  necessary  that  a  thing  should  be,  it  is 
possible  that  it  should  be.  (For  if  not,  the  opposite  follows, 
since  one  or  the  other  must  follow;  so,  if  it  is  not  possible,  it 
is  impossible,  and  it  is  thus  impossible  that  a  thing  should  be 
which  must  necessarily  be;  which  is  absurd.)  Yet  from  the 
proposition  'it  may  be'  it  follows  that  it  is  not  impossible,  and 
from  that  it  follows  that  it  is  not  necessary;  it  comes  about 
therefore  that  the  thing  which  must  necessarily  be  need  not 
be;  which  is  absurd.  But  again,  the  proposition  'it  is  necessary 
that  it  should  be'  does  not  follow  from  the  proposition  'it 
may  be',  nor  does  the  proposition  'it  is  necessary  that  it 
should  not  be'.  .  .  .  For  if  a  thing  may  be,  it  may  also  not  be, 
but  if  it  is  necessary  that  it  should  be  or  that  it  should  not  be, 
one  of  the  two  alternatives  will  be  excluded.  It  remains, 
therefore,  that  the  proposition  'it  is  not  necessary  that  it 
should  not  be'  follows  from  the  proposition  'it  may  be'.  For 
this  is  true  also  of  that  which  must  necessarily  be. 

The  sequence  of  thought  here  is,  in  summary  form:  If  something 
is  necessary,  then  it  is  also  possible;  but  what  is  possible,  can  also 
not  be  (it  is  not  impossible  that  it  should  not  be) ;  but  from  that  it 
follows  that  it  is  not  necessary,  and  so  a  contradiction  results.  The 
solution  consists  in  distinguishing  the  two  meanings  of  being 

15.071  One-sided  possibility:  that  is  possible  which  not 
necessarily  is  not  (which  is  not  impossible). 



15.072  Two-sided  possibility:  that  is  possible  which  neither 
necessarily  is  nor  necessarily  is  not  (nor  impossibly  is). 

This  second,  two-sided  possibility  is  the  one  intended  in  the 
syllogistic,  and  Aristotle  only  uses  the  first  kind  when  forced  to  it. 
He  defines  two-sided  possibility  in  the  Prior  Analytics  thus: 

15.08  I  use  the  terms  'to  be  possible'  and  'the  possible'  of 
that  which  is  not  necessary  but,  being  assumed,  results  in 
nothing  impossible. 

It  coincides,  as  can  be  seen,  with  the  definition  just  fdven  above; 
we  find  a  similar  one  in  the  Metaphysics  (15.09). 

In  two  texts  -  but  both  extremely  unclear  and  so  hard  to  reconcile 
with  the  teaching  as  a  whole  as  to  constitute  an  unsolved  problem 
(15.10)  -  Aristotle  subdivides  two-sided  possibility.  In  the  first 
passage  he  speaks  of  a  possibility  in  the  sense  of  'in  most  cases' 
(&$  £7il  to  7U>Xu)  and  of  another  besides  (15.11);  in  the  second 
passage  he  distinguishes  between  a  'natural'  (to  TC(pi)XO£  \jtA^/zvj 
and  an  indeterminate  (to  aopiaTOv)  or  'contingent'  (to  arco  t's/y^ 
possibility  which,  so  he  says,  is  no  concern  of  science.  Both  passages 
are  probably  interpolations. 


The  normal  use  of  'possible'  in  the  sense  of  two-sided  possibility 
is  a  distinguishing  characteristic  of  Aristotle's  modal  logic.  Another, 
of  no  less  importance,  is  his  view  of  the  structure  of  modal  sentences. 
He  only  gives  explicit  expression  to  this  view  in  one  place,  but  it 
lies  at  the  base  of  the  whole  modal  syllogistic  and  exercises  a  most 
remarkable  influence. 

15.13  The  expression  'it  is  possible  for  this  to  belong  to 
that'  may  be  understood  in  two  senses:  either  as  'to  the  thing 
to  which  that  belongs'  or  as  'to  the  thing  to  which  that  can 
belong' ;  for  'to  that  of  which  B  (is  predicated)  A  can  (belong)' 
means  one  of  the  two:  'to  that  of  which  B  is  predicated'  or 
'to  that  of  which  (B  as)  possibly  (belonging)  is  predicated'. 

This  contains  two  points: 

First  a  sentence  of  the  form  'A  belongs  to  B'  is  paraphrased  by  the 
formula  'to  that  to  which  B  belongs  (of  which  B  is  predicated)  A 
also  belongs' :  implying  a  very  subtle  analysis  of  the  sentence, 
reminiscent  of  the  modern  formal  implication,  which  we  find  else- 
where in  the  Analytics  (cf.  14.24). 

Secondly  it  can  be  gathered  from  this  text  that  the  modal  functor 
does  not  determine  the  sentence  as  a  whole,  but  part  of  it.  So  that 



for  Aristotle  a  modal  sentence  is  not  to  be  conceived  in  such  a  sense  as : 
'It  is  possible  that:  A  belongs  to  B\  The  modal  functor  does  not 
precede  the  whole  sentence  but  one  of  its  arguments.  This  distinction 
quickly  becomes  still  clearer,  for  the  distinction  is  three  times  made 
between  two  possible  cases: 

1.  to  that  to  which  B  belongs,  A  also  can  belong, 

2.  to  that  to  which  B  can  belong,  A  also  can  belong.  In  the 
first  case  the  modal  functor  determines  only  the  consequent,  in  the 
second  case  it  determines  the  antecedent  too. 

This  analysis  is  not  expressly  extended  to  necessity,  but  that 
extension  must  be  supposed :  for  otherwise  many  syllogisms  would  be 


In  the  Hermeneia  Aristotle  establishes  a  'logical'  square  for  senten- 
ces with  modal  functors,  in  which  the  two  expressions  for  'possible' 
(Suvoctov  and  evSe^ofjisvov)  mean  one-sided  possibility.  This  square 
can  be  compressed  into  the  following  scheme,  in  which  all  expressions 
in  any  one  row  are  equivalent: 

possible  not  impossible  not  necessary  not 

not  possible  impossible  necessary  not 

possible  not  not  impossible  not  not  necessary 

not  possible  not  impossible  not  necessary  (15.14) 

More  complicated  is  the  doctrine  of  the  negation  of  sentences 
containing  the  functor  of  two-sided  possibility.  Since  this  has  been 
defined  by  a  conjunction  of  two  sentences,  Aristotle  rightly  deduces, 
on  the  basis  of  the  so-called  de  Morgan  law  (not  to  be  found  in 
him) : 

15.15  If  anyone  then  should  claim  that  because  it  is  not 
possible  for  C  to  belong  to  all  D,  it  necessarily  does  not  belong 
to  some  D,  he  would  make  a  false  assumption:  for  it  does 
belong  to  all  D,  but  because  in  some  cases  it  belongs  neces- 
sarily, therefore  we  say  that  it  is  not  possible  for  it  to  belong 
to  all.  Hence  both  the  propositions  lA  necessarily  belongs  to 
some  B'  and  'A  necessarily  does  not  belong  to  some  B:  are 
opposed  to  the  proposition  lA  may  belong  to  all  B\ 

The  passage  is  not  quite  clear;  but  the  author's  intention  can 
be  formulated : 

15.151  p  is  not  possible,  if  and  only  if,  one  of  the  two,  p  and 
not-p,  is  necessary. 

From  this  it  results  that  the  negation  of  such  a  sentence  issues  as 
an  alternation,  such  as  is  in  no  case  permissible  as  a  premiss  in  an 
Aristotelian  syllogism.  This  prevents  Aristotle  from  using  reduction 
to  the  impossible  in  certain  cases. 



Another  result  which  Aristotle  subtly  deduces  from  his  supposi- 
tions is  his  doctrine  of  the  equivalence  of  affirmative  and  negative 
sentences  when  they  contain  the  functor  under  consideration  : 

15.16  It  results  that  all  premisses  in  the  mode  of  possibility 
are  convertible  into  one  another.  I  mean  not  that  the  affirma- 
tive are  convertible  into  the  negative,  but  that  those  which 
are  affirmative  in  form  admit  of  conversion  by  opposition, 
e.g.  'it  is  possible  to  belong'  may  be  converted  into  'it  is 
possible  not  to  belong';  'it  is  possible  to  belong  to  all'  into  'it 
is  possible  to  belong  to  none'  and  'not  to  all';  'it  is  possible 
to  belong  to  some'  into  'to  some  it  is  possible  not  to  belong'. 
And  similarly  in  other  cases. 

Take  the  three  modal  sentences: 

(a)  'A  possibly  belongs  to  B\ 

(b)  'A  does  not  possibly  belong  to  B\ 

(c)  lA  possibly  does  not  belong  to  B\ 

(b)  is  the  proper  denial  of  (a),  (c)  is  no  denial  of  (a)  but  it  is 
'negative  in  form'.  Then  it  is  stated  that  sentences  such  as  (a) 
imply  those  such  as  (c),  and  are  even  equivalent  to  them.  So  we 
have  following  laws : 

15.161  p  is  possible  if  and  only  if  p  is  not  possible. 

15.162  It  is  possible  that  A  belongs  to  all  B,  if  and  only  if, 
it  is  possible  that  A  belongs  to  no  B. 

15.163  It  is  possible  that  A  belongs  to  some  B,  if  and  only 
if,  it  is  possible  that  to  some  B  A  does  not  belong. 

Law^  analogous  to  those  for  ordinary  conversion  (14.10)  hold 
for  sentences  containing  the  functors  of  necessity  and  one-sided 
possibility  (15.17),  just  parallel  to  the  corresponding  laws  in  asser- 
toric logic  (15.18). 

By  contrast,  the  laws  of  conversion  for  sentences  with  the  two- 
sided  functor  are  different:  the  universal  negative  cannot  be  con- 
verted (15.19),  but  the  particular  negative  can  (15.20).  The  affir- 
mative sentences  are  converted  like  assertoric  ones  (15.21). 


On  this  basis  and  with  the  aid  of  the  same  procedures  developed 
for  the  assertoric  syllogistic,  Aristotle  now  builds  the  vast  structure 
of  his  system  of  syllogisms  with  modal  premisses.  Vast  it  is  even 
in  the  number  of  formulae  explicitly  considered,  they  are  not  fewer 
than  one  hundred  and  thirty  seven.  But  it  appears  much  vaster  -  in 
spite  of  many  points  where  it  is  incomplete  -  in  view  of  the  subtlety 



with  which  the  original  master-logician  operates  in  so  difficult  a 
field.  De  modalibus  non  gustabit  asinus  was  a  medieval  proverb; 
but  one  does  not  need  to  be  a  donkey  to  get  lost  in  this  maze  of 
abstract  laws:  Theophrastus  quite  misunderstood  the  system,  and 
nearly  all  the  moderns,  until  1934. 

The  syllogisms  which  it  comprises  can  be  arranged  in  eight 
groups.  If  we  write  'A7'  for  a  premiss  with  the  functor  'necessary', 
'Af  for  one  with  the  functor  'possible'  and  'A'  for  an  assertoric 
premiss  (that  is,  one  which  predicates  mere  belonging  to),  these 
groups  can  be  shown  as  follows: 

Group  12        3         4  5         6         7         8 

Major  premiss  N       N       A        M         M       A        M       N 

Minor  premiss  N       A        N       M         A        M       N        M 

An.Pr.A,  chap.  8     9-11  9-11  14/17/20  15/18/21      16/19/22 

A  striking  characteristic  of  this  syllogistic  is  that  in  very  many 
syllogisms  the  conclusion  (contrary  to  14.28)  has  a  stronger 
modality  than  the  premisses,  necessity  being  reckoned  as  stronger 
that  mere  belonging  to  and  this  as  stronger  than  possibility. 

15.22  It  happens  sometimes  also  that  when  one  premiss  is 
necessary  the  conclusion  is  necessary,  not  however,  wThen 
either  premiss  is  necessary,  but  only  when  the  major  is,  e.g. 
if  A  is  taken  as  necessarily  belonging  or  not  belonging  to  B, 
but  B  is  taken  as  simply  belonging  to  C:  for  if  the  premisses 
are  taken  in  this  way,  A  will  necessarily  belong  or  not  belong 
to  C.  For  since  A  necessarily  belongs,  or  does  not  belong,  to 
every  B,  and  since  C  is  one  of  the  J5s,  it  is  clear  that  for  C 
also  the  positive  or  the  negative  relation  to  A  will  hold 

And  of  course  that  is  the  case,  if  one  presupposes  the  structure 
of  the  modal  sentences  as  given  above.  For  then  the  syllogism  here 
described  (an  analogue  of  Barbara)  will  be  interpreted  as  follows: 
If  to  all  to  which  B  belongs,  A  necessarily  belongs, 
and  to  that  to  which  C  belongs,  B  belongs, 
then  to  all  to  which  C  belongs,  A  necessarily  belongs, 
which  is  clearly  correct. 

Hence  it  is  quite  wrong  to  extend  the  validity  of  the  principle 
'the  conclusion  follows  the  weaker  premiss'  (cf.  14.28  and  17.17) 
to  Aristotle's  modal  syllogistic. 

Another  striking  fact  is  that  there  are  numerous  valid  modal 
syllogisms  whose  analogues  in  the  assertoric  syllogistic  are  invalid, 
as  for  instance  those  two  negative  premisses  (in  opposition  to 
14.25) ;  this  is  especially  the  case  when  the  modal  syllogism  has  a 
premiss  with  the  functor  of  possibility  where  the  assertoric  analogue 
has  an  affirmative.  For,  as  has  been  said,  affirmative  and  negative 



possible  premisses  are  equivalent  and  can  replace  one  another. 
We  take  as  an  example  a  passage  where  Aristotle,  after  giving  an 
analogue  of  Barbara,  in  the  fourth  group,  to  which  he  refers  by 
such  phrases  as  'previously',  'the  same  syllogism',  'as  before',  then 


15.23  Whenever  A  may  belong  to  all  B,  and  B  may  belong 

to  no  67,  then  indeed  no  syllogism  results  from  the  premisses 
assumed,  but  if  the  premiss  BC  is  converted  after  the  manner 
of  problematic  propositions,  the  same  syllogism  results  as 
before.  For  since  it  is  possible  that  B  should  belong  to  no  C, 
it  is  possible  also  that  it  should  belong  to  all  C.  This  has  been 
stated  above.  Consequently  if  B  is  possible  for  all  C,  and  A 
is  possible  for  all  B,  the  same  syllogism  again  results.  Similarly 
if  in  both  the  premisses  the  negative  is  joined  with  'it  is 
possible':  e.g.  if  A  may  belong  to  none  of  the  Bs,  and  B  to 
none  of  the  Cs.  No  syllogism  results  from  the  assumed  premis- 
ses, but  if  they  are  converted  we  shall  have  the  same  syllogism 
as  before. 

This  syllogistic  is,  like  the  assertoric,  axiomatized.  There  serve 
as  axioms  the  syllogisms  of  the  first  figure  in  all  groups,  except  the 
sixth  and  eighth,  together  with  the  laws  of  conversion  and,  when 
assertoric  premisses  occur,  principles  of  the  assertoric  syllogistic.  The 
other  syllogisms  are  reduced  to  those  axioms,  mostly  by  conversion 
of  premisses  (direct  procedure).  Reduction  to  the  impossible  serves 
to  prove  syllogisms  of  the  first  figure  in  the  eighth  group  and  the 
analogue  of  Bocardo  in  the  fifth.  The  analogues  of  Baroco  and 
Bocardo  in  the  first  group  are  proved  only  by  ecthesis,  while  the 
same  analogues  in  the  second  and  third  groups  remain  unproved, 
though  it  should  not  be  hard  to  prove  them. 

The  hardest  problem  for  Aristotle  are  the  syllogisms  of  the  sixth 
group.   The  first  figure  ones  among  them  rightly  do  not  rank  as 
intuitively  evident;  e.g.  the  analogue  of  Barbara  would  be: 
If  to  all  to  which  B  belongs  A  belongs, 
and  to  all  to  which  C  belongs,  B  may  belong, 
then  to  all  to  which  C  belongs,  A  belongs. 
For  this  to  be  evident  one  would  have  to  see  the  Tightness  of  the 
sentence  'To  all  to  which  B  may  belong,  B  belongs' ;  but  according 
to  the  definition  of  possibility,  that  is  false.  The  details  of  Aristotle's 
complicated    attempts    to   validate   this    syllogism    are    matter    of 
conjecture  and  dispute,   but  the  fact  that  he  has  to  replace  the 
problematic  minor  premiss  with  an  assertoric  one  is  a  sufficient 
indication  of  its  essential  weakness  (15.24).  However,  the  passage 
which  contains  this  'proof  is  one  of  the  few  where  Aristotle  rises  to 



the  use  of  propositional  variables,  and  for  that  reason  remains  of  the 
greatest  logical  interest  (15.25). 

This  abortive  proof  is  moreover  not  the  only  inconsistency  in  the 
Aristotelian  modal  logic.  There  are  for  instance  essential  difficulties 
in  connection  with  the  conversion  of  premisses  with  the  functor  of 
necessity,  and  consequently  in  the  proving  of  many  syllogisms  which 
contain  such  premisses.  In  general  one  gets  the  impression  that  this 
modal  logic,  by  contrast  to  the  assertoric  syllogistic,  is  still  only 
in  a  preliminary  and  incomplete  stage  of  development. 


For  those  reasons  there  is  no  possible  doubt  that  the  theory  of 
what  Aristotle  would  have  called 'analytic'  syllogisms  is  his  chief  ac- 
complishment in  the  field  of  formal  logic.  And  so  great  an  accomplish- 
ment is  it  from  the  historical  and  systematic  points  of  view,  that 
later,  'classical'  logicians  have  mostly  overlooked  all  else  in  his 
work.  Yet  the  Organon  contains  a  profusion  of  laws  and  rules  of 
other  kinds.  Aristotle  himself  recognized  some  of  them  as  autono- 
mous formulae,  irreducible  to  his  syllogistic.  In  other  words:  he 
saw  that  a  'reduction'  of  these  laws  and  rules  to  the  syllogistic  is 
impossible  -  a  thing  which  all  too  many  after  him  did  not  see. 

From  the  historical  standpoint  these  formulae  are  to  be  divided 
into  three  classes :  first  we  have  the  formulae  which  are  to  be  attri- 
buted to  a  period  in  which  Aristotle  had  not  yet  discovered  his 
analytic  syllogisms.  These  are  to  be  found  in  the  Topics  (and  in  the 
Rhetoric).  Some  of  them  were  later  re-edited  with  the  help  of  vari- 
ables, and  recognized  as  valid  also  in  the  period  of  the  Analytics. 
Secondly  there  are  the  formulae  which  Aristotle  indeed  considered 
but  mistakenly,  as  analytic,  the  syllogismi  obliqui  as  they  were  later 
called.  Finally,  in  reviewing  the  completed  system  of  his  syllogistic 
he  discovered  the  'hypothetical'  procedure  and  in  some  cases 
attained  to  full  consciousness  of  propositional  formulae. 

But  all  these  formulae  are  contained  only  in  asides,  and  were 
never  systematically  developed  as  was  the  syllogistic.  Furthermore, 
Aristotle  thought,  quite  rightly  in  view  of  his  methodological 
standpoint,  that  only  the  analytic  formulae  were  genuinely  'scienti- 
fic', i.e.  usable  in  demonstration. 

We  give  first  the  passages  which,  as  it  seems,  introduce  us  to 
Aristotle's  last  thoughts  on  this  question,  then  the  actual  non- 
analytic  formulae  divided  into  five  classes:  those  belonging  to  the 
logic  of  classes,  to  the  theory  of  identity,  to  the  'hypothetical' 
syllogistic,  to  the  theory  of  relations,  and  to  propositional  logic. 



16.01  In  some  arguments  it  is  easy  to  see  what  is  wanting, 
but  some  escape  us,  and  appear  to  be  syllogisms,  because 
something  necessary  results  from  what  has  been  laid  down, 
e.g.  if  the  assumptions  were  made  that  substance  is  not 
annihilated  by  the  annihilation  of  what  is  not  substance,  and 
that  if  the  elements  out  of  which  a  thing  is  made  are  annihi- 
lated, then  that  which  is  made  out  of  them  is  destroyed :  these 
propositions  being  laid  down,  it  is  necessary  that  any  part 
of  substance  is  substance;  this  has  not,  however,  been  drawn 
by  syllogism  from  the  propositions  assumed,  but  premisses 
are  wanting.  Again  if  it  is  necessary  that  animal  should 
exist,  if  man  does,  and  that  substance  should  exist  if  man 
does:  but  as  yet  the  conclusion  has  not  been  drawn  syllo- 
gistically:  for  the  premisses  are  not  in  the  shape  we  required. 
We  are  deceived  in  such  cases  because  something  necessary 
results  from  what  is  assumed,  since  the  syllogism  also  is 
necessary.  But  that  which  is  necessary  is  wider  than  the 
syllogism :  for  every  syllogism  is  necessary,  but  not  every- 
thing which  is  necessary  is  a  syllogism. 

We  must  pass  over  the  first  example,  about  parts  of  substance,  as 
its  elucidation  would  take  up  too  much  space.  But  the  second  is 
clear;  it  concerns  a  law,  not  of  propositional,  but  of  predicate  logic: 

16.011  If,  when  x  is  A  then  it  is  B,  and  when  x  is  B  then 
it  is  C,  then,  when  x  is  A  then  it  is  C. 

This  is  a  correct  logical  formula,  and  Aristotle  is  quite  right  in 
saying  that  it  permits  necessary  inference.  Hence  he  also  realized 
that  it  falls  under  his  definition  of  syllogism  (10.05).  But  he  refuses 
to  admit  it  as  syllogism.  That  means  that  his  conception  of  syllo- 
gism had  developed  between  the  time  when  he  penned  the  definition 
and  that  when  he  penned  this  passage.  The  definition  applies  to  all 
correct  logical  formulae  (and  substitutions  in  them),  but  only  a 
sub-class  retains  the  name  'syllogism'.  We  know  what  this  sub-class 
is  that  of  the  'analytic'  syllogisms.  All  other  formulae  may  indeed  be 
logically  necessary,  but  are  not  genuine  syllogisms. 

This  distinction  is  not  merely  a  matter  of  terminology.  That 
becomes  evident  in  the  passages  where  Aristotle  deals  with  the 
'hypothetical'  syllogisms. 

16.02  We  must  not  try  to  reduce  hypothetical  syllogisms; 
for  with  the  given  premisses  it  is  not  possible  to  reduce  them. 
For  they  have  not  been  proved  by  syllogism,  but  assented  to 



by  agreement.  For  instance  if  a  man  should  suppose  that  unless 
there  is  one  faculty  of  contraries,  there  cannot  be  one  science, 
and  should  then  argue  that  there  is  no*  (one)  faculty  of 
contraries,  e.g.  of  what  is  healthy  and  what  is  sickly:  for  the 
same  thing  will  then  be  at  the  same  time  healthy  and  sickly. 
He  has  shown  that  there  is  not  one  faculty  of  all  contraries, 
but  he  has  not  proved  that  there  is  not  a  science.  And  yet 
one  must  agree.  But  the  agreement  does  not  come  from  a 
syllogism,  but  from  an  hypothesis.  This  argument  cannot  be 
reduced :  but  the  proof  that  there  is  not  a  single  faculty  can. 

Evidently  a  substitution  is  there  being  made  in  the  law: 
If  (1)  when  not  p  then  not  q,  and  (2)  not  p,  then  (3)  not  q.  (2)  is 
proved  by  an  analytic  syllogism,  but  as  (1)  is  merely  supposed  and 
not  proved,  the  conclusion  (3)  also  counts  as  not  proved.  That  may 
be  so,  but  Aristotle  has  not  noticed  that  the  assumed  formula  is  no 
mere  supposition  but  a  correct  logical  law.  The  matter  is  still  worse 
in  the  next  text,  an  immediate  continuation  of  the  last: 

16.03  The  same  holds  good  of  arguments  which  are  brought 
to  a  conclusion  per  impossibile.  These  cannot  be  analyzed 
either;  but  the  reduction  to  what  is  impossible  can  be  analyzed 
since  it  is  proved  by  syllogism,  though  the  rest  of  the  argument 
cannot,  because  the  conclusion  is  reached  from  an  hypothesis. 
But  these  differ  from  previous  arguments :  for  in  the  former  a 
preliminary  agreement  must  be  reached  if  one  is  to  accept 
the  conclusion;  e.g.  an  agreement  that  if  there  is  proved  to 
be  one  faculty  of  contraries,  then  contraries  fall  under  the 
same  science;  whereas  in  the  latter,  even  if  no  preliminary 
agreement  has  been  made,  men  still  accept  the  reasoning, 
because  the  falsity  is  patent,  e.g.  the  falsity  of  what  follows 
from  the  assumption  that  the  diagonal  is  commensurate,  viz. 
that  then  odd  numbers  are  equal  to  evens. 

So  in  reduction  to  the  impossible  too,  Aristotle  regards  the 
inference  as  not  'demonstrated',  though  he  has  to  recognize  that  no 
agreement  needs  to  be  presupposed  to  warrant  inference. 

One  could  express  this  doctrine  as  follows:  the  class  of  correct 
formulae  contains  two  sub-classes :  that  of  the  'better'  and  that  of  the 
'less  good'  in  relation  to  'scientific  demonstration'.  The  less  good 

*  Reading  uia  with  the  manuscript  tradition  A2B2C2T,  against  the  (better) 
tradition  ABCnAl,  Waitz  and  Ross.  For  raccra,  read  by  the  latter,  would  mean 
a  logical  mistake  in  Aristotle  which  seems  to  me  unlikely  in  this  connection.  For 
the  critical  apparatus  vid.  Ross  ad  loc. 



are  precisely  our  non-analytic  formulae,  for  which  we  have  chosen 
this  name  because  according  to  Aristotle  they  are  not  reducible  to 
the  classical  syllogistic,  'not  analyzable  into  the  figures'.  (In  this  lie 
is  evidently  right,  by  contrast  to  a  certain  tradition.)  That  does  not 
mean  that  these  formulae  are  worthless  for  him;  on  the  contrary 
he  views  them  with  a  lively  interest. 

16.04  These  points  will  be  made  clearer  by  the  sequel, 
when  we  discuss  the  reduction  to  impossibility.  ...  In  the 
other  hypothetical  syllogisms,  I  mean  those  which  proceed 
by  substitution,  or  positing  a  certain  quality,  the  inquiry  will 
be  directed  to  the  terms  of  the  problem  to  be  proved  -  not  the 
terms  of  the  original  problem,  but  the  new  terms  introduced; 
and  the  method  of  the  inquiry  will  be  the  same  as  before. 
But  we  must  consider  and  determine  in  how  many  ways 
hypothetical  syllogisms  are  possible. 

16.05  Many  other  arguments  are  brought  to  a  conclusion 
by  the  help  of  an  hypothesis;  these  we  ought  to  consider  and 
mark  out  clearly.  We  shall  describe  in  the  sequel  differences, 
and  the  various  ways  in  which  hypothetical  arguments  are 
formed:  but  at  present  this  much  must  be  clear,  that  it  is  not 
possible  to  resolve  such  arguments  into  the  figures.  And  we 
have  explained  the  reason. 

On  that  Alexander  of  Aphrodisias  remarks: 

16.06  He  says  that  many  others  (syllogisms)  besides  are 
formed  from  hypotheses,  and  promises  to  treat  thoroughly 
of  them  later.  But  no  writing  of  his  on  this  subject  is  extant. 


16.07  If  man  (is)  an  animal,  what  is  not-animal  is  not  man. 

16.08  If  the  pleasant  (is)  good,  the  not-good  (is)  not  pleasant. 

Notice  that  quantifiers  are  here  lacking:  so  it  is  not  a  question  of 
contraposition  in  the  ordinary  sense  of  the  word. 

Aristotle  was  well  aware  that  conversion  of  such  sentences  is 
invalid : 

16.09  For  animal  follows  on  man,  but  not-animal  does  not 
(follow)  on  not-man;  the  reverse  is  the  case. 

Here  there  belong  perhaps  some  rules  which  otherwise  interpreted 
could  be  counted  in  with  those  of  the  'logical  square' : 



16.10  When  we  have  shown  that  a  predicate  belongs  in 
every  case,  we  shall  also  have  shown  that  it  belongs  in  some 
cases.  Likewise,  also,  if  we  show  that  it  does  not  belong  in 
any  case,  we  shall  also  have  shown  that  it  does  not  belong  in 
every  case. 

It  is  to  be  noted  here  that  these  formulae  are  not  laws  but,  as  is 
mostly  the  case  in  the  Topics,  rules. 

A  series  of  similar  laws  is  concerned  with  contrariety  (in  the  sense 
of  the  earlier  notion:  12.02): 

16.11  Health  follows  upon  vigour,  but  disease  does  not 
follow  upon  debility;  rather  debility  follows  upon  disease. 

16.12  Public  opinion  grants  alike  the  claim  that  if  all 
pleasure  be  good,  then  also  all  pain  is  evil,  and  the  claim  that 
if  some  pleasure  be  good,  then  also  some  pain  is  evil.  More- 
over, if  some-  form  of  sensation  be  not  a  capacity,  then  also 
some  form  of  failure  of  sensation  is  not  a  failure  of  capa- 
city. .  .  .  Again,  if  what  is  unjust  be  in  some  cases  good,  then 
also  what  is  just  is  in  some  cases  evil;  and  if  what  happens 
justly  is  in  some  cases  evil,  then  also  what  happens  unjustly 
is  in  some  cases  good. 

It  may  be  doubted  whether  Aristotle  continued  to  recognize  these 
laws  as  valid.  But  they  are  not  without  historical  and  even  syste- 
matic interest. 


As  already  noted  (11.11)  Aristotle  distinguishes  three  kinds  of 
identity.  Concerning  the  first,  numerical  identity,  he  developed 
the  outline  of  a  theory;  its  discovery  is  often  falsely  attributed  to 

16.13  Again,  look  and  see  if,  supposing  the  one  to  be  the 
same  as  something,  the  other  also  is  the  same  as  it:  for  if 
they  be  not  both  the  same  as  the  same  thing,  clearly  neither 
are  they  the  same  as  one  another.  Moreover,  examine  them  in 
the  light  of  their  accidents  or  of  the  things  of  which  they  are 
accidents:  for  any  accident  belonging  to  the  one  must  belong 
also  to  the  other,  and  if  the  one  belongs  to  anything  as  an 
accident,  so  must  the  other  also.  If  in  any  of  these  respects 
there  is  a  discrepancy,  clearly  they  are  not  the  same. 

We  have  here  in  very  compressed  form  a  rather  highly  developed 
doctrine  of  identity;  indeed  this  text  contains  a  greater  number  of 



fundamental  laws  of  identity  than  the  corresponding  chapter  of 
Principia  Malhemalica  (*13),  and  moreover  Aristotle  was  the  first 
to  call  to  mind  identity,  in  the  passage  mentioned  above  (\2AHt.  The 
laws  here  sketched,  can  be  formulated  as  follows  with  the  help  of 

16.131  If  B  is  identical  with  A,  and  C  is  not  identical  with 
A,  then  B  and  C  are  not  identical. 

16.132  If  A  and  B  are  identical,  then  (for  all  C):  if  Cbelongs 

to  A,  then  it  belongs  also  to  B. 

16.133  If  A  and  B  are  identical,  then  (for  all  C) :  if  A  belongs 
to  C,  then  it  belongs  also  to  B. 

16.134  If  there  is  a  C  which  belongs  to  A  but  not  to  B,  then 
A  and  B  are  not  identical. 

16.135  If  there  is  a  C  to  which  A  belongs  but  B  does  not, 
then  A  and  B  are  not  identical. 

Admittedly  the  last  two  laws  are  only  hinted  at.  In  another 
passage  we  find : 

16.14  For  only  to  things  that  are  indistinguishable  and  one 
in  essence  is  it  generally  agreed  that  all  the  same  attributes 

This  is  almost  the  Leibnizian  principium  indiscernibilium  in  so 
many  words,  originating  as  we  see  with  Aristotle.  It  is  remarkable 
that  we  do  not  find  the  simple  principle : 

16.141  If  A  is  identical  with  B,  and  B  with  C,  then  A  is 
identical  with  C. 


Aristotle  did  not  know  the  expression  'hypothetical  syllogism', 
but  he  often  speaks  of  syllogisms  from  hypotheses  (si*  \)~o$£gz(x>c,) . 
We  have  shown  above  (in  the  commentary  on  citations  16.02  and 
16.03)  that  in  general  these  need  not  be  hypotheses;  usually  it  is  a 
matter  only  of  logical  laws  or  rules,  similar  to  syllogisms  in  certain 
respects  but  not  reducible  to  them.  We  have  already  seen  some 
examples  of  such  formulae.  Here  are  some  more  which  Aristotle 
would  probably  class  with  them. 

16.15  The  refutation  which  depends  upon  the  consequent 
arises  because  people  suppose  that  the  relation  of  consequence 
is  convertible.  For  whenever,  if  this  is,  that  necessarily  is,  they 
suppose  that  also  when  that  is,  this  is. 



16.16  When  two  things  are  so  related  to  one  another,  that 
if  the  one  is,  the  other  necessarily  is,  then  if  the  latter  is  not, 
the  former  will  not  be  either,  but  if  the  latter  is,  it  is  not 
necessary  that  the  former  should  be. 

16.17  If  this  follows  that,  it  is  claimed  that  the  opposite 
of  this  will  follow  the  opposite  of  that. . . .  But  that  is  not  so; 
for  the  sequence  is  vice  versa. 

16.18  In  regard  to  subjects  which  must  have  one  and  one 
only  of  two  predicates,  as  (e.g.)  a  man  must  have  either 
disease  or  health,  supposing  we  are  well  supplied  as  regards 
the  one  for  arguing  its  presence  or  absence,  we  shall  be  well 
equipped  as  regards  the  remaining  one  as  well.  This  rule  is 
convertible  for  both  purposes :  for  when  we  have  shown  that 
the  one  attribute  belongs,  we  shall  have  shown  that  the 
remaining  one  does  not  belong;  while  if  we  show  that  the  one 
does  not  belong,  we  shall  have  shown  that  the  remaining  one 
does  belong. 

We  have  there,  evidently,  the  exclusive  alternative  (negation  of 

16.19  In  general  whenever  A  and  B  are  such  that  they 
cannot  belong  at  the  same  time  to  the  same  thing,  and  one 
of  the  two  necessarily  belongs  to  everything,  and  again  C  and 
D  are  related  in  the  same  way,  and  A  follows  C  but  the 
relation  cannot  be  reversed,  then  D  must  follow  B  and  the 
relation  cannot  be  reversed.  And  A  and  D  may  belong  to  the 
same  thing,  but  B  and  C  cannot.  First  it  is  clear  from  the 
following  consideration  that  D  follows  B.  For  since  either  C 
or  D  necessarily  belongs  to  everything;  and  since  C  cannot 
belong  to  that  to  which  B  belongs,  because  it  carries  A  along 
with  it  and  A  and  B  cannot  belong  to  the  same  thing;  it  is 
clear  that  D  must  follow  B.  Again  since  C  does  not  reciprocate 
with  A,  but  C  or  D  belongs  to  everything,  it  is  possible  that  A 
and  D  should  belong  to  the  same  thing.  But  B  and  C  cannot 
belong  to  the  same  thing,  because  A  follows  C;  and  so  some- 
thing impossible  results.  It  is  clear  then  that  B  does  not 
reciprocate  with  D  either,  since  it  is  possible  that  D  and  A 
should  belong  at  the  same  time  to  the  same  thing. 

This  text  is  one  of  the  peaks  of  Aristotelian  logic:  the  founder  of 
our  science  conducts  himself  with  the  same  sureness  and  freedom 
as  in  the  best  parts  of  his  syllogistic,  though  here  dealing  with  a  new 



field,  that  of  non-analytic  formulae.  The  run  of  this  text  can   be 
formulated  thus: 

(1)  For  all  X:  A  or  B  (and  not  both)  belongs  to  X,  and 

(2)  for  all  X:  C  or  D  (and  not  both)  belongs  to  X,  and 

(3)  for  all  X:  if  C  belongs  to  X,  then  it  belongs  also  to  A. 
From  these  hypotheses  there  follows  on  the  one  hand : 

(4)  for  all  X:  if  B  belongs  to  X,  then  it  belongs  also  to  D, 
and  on  the  other: 

(5)  for  all  X:  not  both  B  and  C  belong  to  X. 

These  consequences  are  quite  correct.  The  thing  to  notice  is  that 
there  are  three  different  binary  propositional  functors  ('or',  'and',  'if... 
then').  In  his  proof  Aristotle  uses,  among  others,  the  following  three 
laws,  apparently  with  full  consciousness: 

16.191  For  all  X:  if  not  both  A  and  B  belong  to  X,  and  B 
belongs  to  X,  then  A  does  not  belong  to  it. 

16.192  For  all  X :  if,  when  A  belongs  to  X  B  also  belongs  to 
X,  but  B  does  not  belong  to  X,  then  A  also  does  not  belong 

16.193  For  all  X:  if  either  A  or  B  belongs  to  X,  and  A  does 
not  belong  to  it,  then  B  does  belong  to  it. 


16.20  If  knowledge  be  a  conceiving,  then  also  the  object 
of  knowledge  is  an  object  of  conception. 

16.21  If  the  object  of  conception  is  in  some  cases  an  object 
of  knowledge,  then  also  some  form  of  conceiving  is  knowledge. 

16.22  If  pleasure  is  good,  then  too  a  greater  pleasure  is  a 
greater  good ;  and  if  injustice  is  bad,  then  too  a  greater  injustice 
is  a  greater  evil. 

In  this  connection  the  following  piece  of  history  deserves  to  go  on 
record.  De  Morgan  stated  that  the  whole  Aristotelian  logic  was 
unable  to  prove  that  if  the  horse  is  an  animal,  then  the  head  of  the 
horse  is  head  of  an  animal.  The  reproach  is  evidently  unjustified, 
since  the  law  stated  in  16.20  is  just  what  is  needed  for  this  proof. 
Further,  Whitehead  and  Russell  (16.23)  remark  that  the  supposed 
lack  of  this  law  is  really  a  good  point  abou£  Aristotelian  logic,  since 
it  is  invalid  without  an  additional  existential  postulate.  This  may 
be  right  in  relation  to  De  Morgan's  problem,  i.e.  if  he  understood 
'horse'  as  an  individual  name;  but  the  law  in  which  16.20  is  a 
substitution  is  correct  -  since  it  concerns  not  an  individual  but  a 
class  name  ('knowledge'). 



Aristotle  gives  three  further  laws  of  the  logic  of  relations  in  the 
chapter  about  those  syllogisms  which  later  came  to  be  called 
'obliqui' : 

16.24  That  the  first  term  belongs  to  the  middle,  and  the 
middle  to  the  extreme,  must  not  be  understood  in  the  sense 
that  they  can  always  be  predicated  of  one  another.  .  .  .  But 
we  must  suppose  the  verb  'to  belong'  to  have  as  many 
meanings  as  the  senses  in  which  the  verb  'to  be'  is  used,  and 
in  which  the  assertion  that  a  thing  'is'  may  be  said  to  be  true. 
Take  for  example  the  statement  that  there  is  a  single  science 
of  contraries.  Let  A  stand  for  'there  being  a  single  science', 
and  B  for  things  which  are  contrary  to  one  another.  Then  A 
belongs  to  jB,  not  in  the  sense  that  contraries  are  the  fact  of 
there  being  a  single  science  of  them,  but  in  the  sense  that  it  is 
true  to  say  of  the  contraries  that  there  is  a  single  science  of 

16.25  It  happens  sometimes  that  the  first  term  is  stated  of 
the  middle,  but  the  middle  is  not  stated  of  the  third  term,  e.g. 
if  wisdom  is  knowledge,  and  wisdom  is  of  the  good,  the  con- 
clusion is  that  there  is  knowledge  of  the  good.  The  good  then 
is  not  knowledge,  though  wisdom  is  knowledge. 

16.26  Sometimes  the  middle  term  is  stated  of  the  third,  but 
the  first  is  not  stated  of  the  middle,  e.g.  if  there  is  a  science  of 
everything  that  has  a  quality,  or  is  a  contrary,  and  the  good 
both  is  a  contrary  and  has  a  quality,  the  conclusion  is  that 
there  is  a  science  of  the  good,  but  the  good  is  not  science,  nor 
is  that  which  has  a  quality  or  is  a  contrary,  though  the  good  is 
both  of  these. 

16.27  Sometimes  neither  the  first  term  is  stated  of  the 
middle,  nor  the  middle  of  the  third,  while  the  first  is  sometimes 
stated  of  the  third,  and  sometimes  not:  e.g.  if  there  is  a  genus 
of  that  of  which  there  is  a  science,  and  if  there  is  a  science  of 
the  good,  we  conclude  that  there  is  a  genus  of  the  good.  But 
nothing  is  (there)  predicated  of  anything.  And  if  that  of  which 
there  is  a  science  is  a  genus,  and  if  there  is  a  science  of  the  good, 
we  conclude  that  the  good  is  a  genus.  The  first  term  then  is 
predicated  of  the  extreme,  but  in  the  premisses  one  thing  is 
not  stated  of  another. 

We  have  here  four  more  relational  laws ;  of  greater  importance  is  the 
introductory  remark  that  the  so-called  'copula'  need  not  be  'is'  but 
can  be  replaced  by  some  other  relation.  A  further  interesting  fact  is 



that  Aristotle  presupposes  inter  alia  the  following  law  from  the  logic 
of  classes  (in  16.26): 

16.261   For  all  x:  if  x  is  A  and  /J,  then  x  is  A  or  B. 

The  introductory  remark  admittedly  only  reveals  an  intuition 
that  is  undeveloped.  Nor  did  Aristotle  link  it  up  with  his  own  pene- 
trating thesis  of  the  manifold  structure  of  the  sentence  according  to 
the  diversity  of  the  categories  (11.15),  so  rising  to  a  higher  syste- 
matic unity.  Nevertheless  the  text  cited  does  contain  the  beginnings 
of  a  logic  of  relations. 

Finally  we  can  collect  from  at  least  six  places  in  the  Topics  a 
group  of  rules,  totalling  eighteen  altogether,  which  perhaps  are 
to  be  interpreted  as  belonging  to  the  logic  of  relations.  We  give  three 
of  them,  again  concerned  with  'more' : 

16.28  Moreover,  argue  from  greater  and  less  degrees.  In 
regard  to  greater  degrees  there  are  four  commonplace  rules. 
One  is :  See  whether  a  greater  degree  of  the  predicate  follows  a 
greater  degree  of  the  subject:  e.g.  if  pleasure  be  good,  see 
whether  also  a  greater  pleasure  be  a  greater  good.  . . .  Another 
rule  is:  If  one  predicate  be  attributed  to  two  subjects;  then 
supposing  it  does  not  belong  to  the  subject  to  which  it  is  the 
more  likely  to  belong,  neither  does  it  belong  where  it  is  less 
likely  to  belong;  while  if  it  does  belong  where  it  is  less  likely 
to  belong,  then  it  belongs  as  well  where  it  is  more  likely.  .  .  . 
Moreover:  If  two  predicates  be  attributed  to  two  subjects, 
then  if  the  one  which  is  more  usually  thought  to  belong  to  the 
one  subject  does  not  belong,  neither  does  the  remaining 
predicate  belong  to  the  remaining  subject;  or.  if  the  one  which 
is  less  usually  thought  to  belong  to  the  one  subject  does  be- 
long, so  too  does  the  remaining  predicate  to  the  remaining 


Finally  we  find  in  Aristotle  four  formulae  belonging  to  the  most 
abstract  part  of  logic,  namely,  propositional  logic.  Two  of  them  even 
contain  propositional  variables: 

16.29  If  when  A  is,  B  must  be,  (then)  when  B  is  not.  A 
cannot  be. 

That  these  are  propositional  variables,  Aristotle  says  expressly: 

16.30  A  is  posited  as  one  thing,  being  two  premisses  taken 



16.31  If,  when  A  is,  B  must  be,  (then)  also  when  A  is 
possible,  B  must  be  possible. 

It  is  to  be  noted  that  these  propositional  variables  permit  sub- 
stitution only  of  quite  determinate  expressions,  namely  conjunctions 
of  premisses  suitable  for  an  analytic  syllogism. 

16.32  From  true  premisses  it  is  not  possible  to  draw  a  false 
conclusion,  but  a  true  conclusion  may  follow  from  false 
premisses,  true,  however,  only  in  respect  to  the  fact,  not  to  the 

That  is  not  yet  the  scholastic  principle  ex  falso  sequitur  quodlibel, 
but  only  the  assertion  that  one  can  form  syllogisms  in  which  one  or 
both  premisses  are  false,  the  conclusion  true. 

16.33  If  the  conclusion  is  false,  the  premisses  of  the  argument 
(Xoyo?)  must  be  false,  either  all  or  some  of  them. 

This  rule  underlies  the  indirect  proof  of  syllogisms  (cf.  14.201-202). 
Note  that  it  is  a  rule,  not  a  law,  and  formulated  quite  generally, 
without  being  limited  to  two  premisses. 


Reviewing  the  logical  doctrines  of  Aristotle  as  presented,  we  can 
state : 

1 .  Aristotle  created  formal  logic.  For  the  first  time  in  history  we 
find  in  him:  (a)  a  clear  idea  of  universally  valid  logic  law,  though  he 
never  gave  a  definition  of  it,  (b)  the  use  of  variables,  (c)  sentential 
forms  which  besides  variables  contain  only  logical  constants. 

2.  Aristotle  constructed  the  first  system  of  formal  logic  that  we 
know.  This  consists  exclusively  of  logical  laws,  and  was  developed 
axiomatically,  even  in  more  than  one  way. 

3.  Aristotle's  masterpiece  in  formal  logic  is  his  syllogistic.  This  is  a 
system  of  term-logic  consisting  of  laws,  not  rules.  In  spite  of  certain 
weaknesses  it  constitutes  a  faultlessly  constructed  system. 

4.  Besides  the  syllogistic,  Aristotle  constructed  other  portions  of 
term-logic,  including  an  extremely  complex  modal  logic,  as  well  as  a 
number  of  laws  and  rules  which  overstep  the  bounds  of  the  syllo- 

5.  At  the  end  of  his  life  Aristotle,  in  a  few  texts,  succeeded  in 
formulating  even  propositional  formulae;  but  these,  like  the  non- 
analytic  formulae  of  term-logic,  he  did  not  develop  systematically. 

6.  Aristotelian  logic,  though  formal,  is  not  formalislic.  It  is  lacking 
also  in  understanding  of  the  difference  between  rules  and  laws,  and 
the  semantics  remain  rudimentary,  in  spite  of  the  many  works 
which  Aristotle  devoted  to  the  subject. 



It  is  no  exaggeration  to  say  that  nothing  comparable  h;j.s  been  seen 
in  the  whole  history  of  formal  logic.  Not  only  is  Aristotle's  logic, 
according  to  all  our  information,  a  completely  new  creation,  but  it 
has  been  brought  even  by  him  to  a  high  degree  of  completeness. 
Since  moreover  Aristotle's  most  important  writings  -  most  impor- 
tant because  they  were  the  only  complete  logical  works  -  survived 
the  cultural  catastrophe  of  Greece,  it  is  no  wonder  that  the  huge  body 
of  doctrine  they  contain  should  have  continued  to  fascinate  nearly 
all  logicians  for  more  than  two  thousand  years,  and  that  the  whole 
history  of  logic  has  developed  along  the  lines  traced  out  in  advance 
by  Aristotle's  thought. 

That  has  not  been  harmless  for  the  development  of  our  science. 
Even  in  antiquity  there  was  a  school  of  logicians  which  introduced  a 
new  set  of  problems  different  from  those  posed  by  the  logic  of  Ari- 
stotle. We  have  only  fragments  of  their  work,  and  the  authority  of 
the  founder  of  logic  was  so  great  that  the  achievements  of  this 
school  were  not  at  all  understood  during  the  long  period  from  the  time 
of  the  Renaissance  up  to  and  including  the  nineteenth  century.  We 
must  now  concern  ourselves  with  them,  but  first  a  brief  word  must 
be  said  about  Aristotle's  first  disciple,  Theophrastus. 


Theophrastus  of  Eresos,  Aristotle's  chief  disciple  and  leader  of  the 
Peripatetic  school  after  the  founder's  death,  has,  in  company  with 
his  less  significant  colleague  Eudemus,  an  important  place  in  the 
history  of  logic,  and  that  in  three  respects.  First,  he  developed  various 
of  his  master's  doctrines  in  such  a  way  as  to  prepare  the  ground  for 
the  later  'classical'  logic;  secondly,  he  set  his  own  quite  different 
system  in  opposition  to  the  Aristotelian  modal  iogic;  thirdly,  he 
developed  a  doctrine  of  hypothetical  arguments  which  was  a  pre- 
paration for  Megarian-Stoic  logic. 

His  very  numerous  works  (17.01)  have  all  perished  save  for  some 
one  hundred  fragments.  These,  however,  are  enough  to  tell  us  that  he 
commented  on  the  most  important  of  Aristotle's  logical  works  ( 1 7.02 1 , 
and  they  give  us  some  insight  into  his  own  logical  thought. 


17.03  Speech  having  a  twofold  relation  -  as  the  philoso- 
pher Theophrastus  has  shown  -  one  to  the  hearers,  to  whom  it 
signifies  something,  the  other  to  the  things  about  which  it 
informs  the  hearers,  there  arise  in  respect  of  the  relation  to  the 
hearers  poetics  and  rhetoric,  ...  in  respect  of  that  to  the  things, 



it  will  be  primarily  the  philosopher's  business,  as  he  refutes 
falsehood  and  demonstrates  truth. 

We  can  see  that  this  is  a  new  semiotic,  with  stress  on  what  is  now 
called  the  'pragmatic'  dimension  of  signs. 

17.04  Theophrastus  rightly  calls  the  singular  sentence 
determined,  the  particular  undetermined. 

17.05  Alexander  opines  that  'not  belonging  to  all'  and  'to 
some  not  belonging'  differ  only  in  the  expression,  whereas 
Theophrastus's  view  is  that  they  differ  also  in  meaning:  for 
'not  belonging  to  all'  shows  that  (something)  belongs  to 
several,  'to  some  not  belonging'  that  (not  belonging)  to  one. 

A  more  important  thought  is  the  following: 

17.06  Consequently  Theophrastus  says  that  in  some  cases, 
if  the  determination  (of  quantity)  TupoaSiopLCT^o^)  does  not 
also  stand  with  the  predicate,  opposites,  contradictories,  will 
be  true,  e.g.  he  says  that  'Phanias  possesses  knowledge', 
'Phanias  does  not  possess  knowledge',  can  both  be  true. 

This  is  not  a  matter,  as  Theophrastus  mistakenly  supposed,  of 
quantification  of  the  predicate,  which  Aristotle  had  rejected  (12.03), 
but  of  a  quantification  of  both  parts  of  a  subject  when  there  is  a 
two-place  functor  (cf.  44.22  ff.).  This  structure  was  only  later  treated 
in  detail  (cf.  28.15 ff.,  42.06,  42.22).  We  have  here  the  first  beginnings 
of  it. 

17.07  In  those  premisses  which  potentially  contain  three 
terms,  viz.  those  .  .  .  which  Theophrastus  called  xa-ra  7rpocrAY)tJnv 
(for  these  have  three  terms  in  a  sense;  since  in  (the  premiss)  'to 
all  of  that  to  all  of  which  B  belongs,  A  also  belongs'  in  the  two 
terms  A  and  B  which  are  explicit  there  is  somehow  comprised 
the  third  of  which  B  is  predicated  .  .  .):  (these  premisses) 
.  .  .  seem  to  differ  from  categorical  ones  only  in  expression,  as 
Theophrastus  showed  in  his  On  Affirmation. 

17.08  But  Theophrastus  in  On  Affirmation  treats  'to  that 
to  which  B  (belongs,  there  belongs  also)  A'  as  equivalent 
(feov  Suva(X£V7]v)  to  'to  all  of  that  to  all  of  which  B  belongs, 
A  (also  belongs)  (cf.  14.24). 

17.09  But  Theophrastus  and  Eudemus  have  given  a 
simpler  proof  that  universal  negative  premisses  can  be  con- 
verted. .  .  .  They  conduct  the  proof  so:  A  belongs  to  no  B. 
If  it  belongs  to  none,  A  must  be  disjoined  (dbus^suxToci)  and 



separated  (xs/wptdTat)  from  B.  But  what  is  disjoined  is 
disjoined  from  something  disjoined.  Therefore  B  too  is  quite 
disjoined  from  A.  And  if  this  is  so,  it  belongs  to  no  A. 

This  shows  that  Theophrastus  takes  a  purely  extensional  view  of 
the  terms  (cf.  §36,  E)  -  so  much  so  that  one  is  led  by  this  text  (as 
by  17.13)  to  think  of  a  diagrammatic  scheme  such  as  Leibniz  used 


17.10  To  these  four  (Aristotelian  syllogisms  of  the  first 
figure)  Theophrastus  added  five  others,  which  are  neither 
perfect  nor  indemonstrable. 

We  no  longer  have  the  relevant  text.  Alexander's  explanations 
(17.11)  show  that  these  are  the  five: 

17.111  If  A  belongs  to  all  B  and  B  to  all  C,  then  too  C 
belongs  to  some  A  (Baralipton). 

17.112  If  A  belongs  to  no  B,  and  B  to  all  C,  then  too  C 
belongs  to  no  A  (Celanles). 

17.1 13  If  A  b  longs  to  all  B  and  B  to  some  C,  then  too  C  be- 
longs to  some  A  (Dabitis). 

17.114  If  A  belongs  to  all  C  and  B  to  no  C,  then  to  some 
A  C  does  not  belong  (Fapesmo). 

17.115  If  A  belongs  to  some  B  and  B  to  no  67,  then  too  to 
some  A  C  does  not  belong  (Frisemomorum). 

These  are  what  were  later  called  the  'indirect'  syllogisms  of  the 
first  figure,  deduced  by  means  of  the  Aristotelian  rules  (cf.  §  13,  D). 


All  the  texts  so  far  quoted  contain  developments  of  or  -  often 
questionable  -  improvements  on  the  Aristotelian  logic.  The  Theo- 
phrastan  theory  of  modal  syllogisms,  on  the  other  hand,  is  an 
entirely  new  system,  set  in  fundamental  oppositions,  as  it  appears, 
to  that  of  Aristotle. 

17.12  Hence  Aristotle  says  that  universal  negative  possible 
premisses  are  not  convertible.  But  Theophrastus  says  that 
these  too,  like  the  other  negatives,  can  be  converted. 

17.13  But  Theophrastus  and  Eudemus,  as  we  have  already- 
explained  at  the  beginning,  say  that  universal  negative 
(possible  premisses)  can  be  converted,  like  universal  negative 
assertoric  and  necessary  ones.  Their  convertibility  they  prove 



thus:  if  A  possibly  does  not  belong  to  all  B,  B  also  possibly 
does  not  (belong)  to  all  A;  for  if  A  possibly  does  not  belong 
to  all  B  .  .  .,  then  A  can  be  disjoined  from  all  B;  but  if  this  is 
so,  B  also  can  be  disjoined  from  A;  and  in  that  case  B  also 
possibly  does  not  belong  to  all  A. 

17.14  It  is  (according  to  Aristotle)  a  property  of  the 
possible  to  convert,  i.e.  the  affirmations  and  negations  con- 
cerning it  follow  on  each  other  .  .  .  but  it  should  be  known  that 
this  conversion  of  premisses  is  not  valid  in  the  school  of 
Theophrastus,  and  they  do  not  use  it.  For  there  is  the  same 
reason  (1)  for  saying  that  the  universal  negative  possible 
(premiss)  is  convertible,  like  the  assertoric  and  necessary,  and 
(2)  (for  saying)  that  affirmative  possibles  are  not  convertible 
into  negatives. 

In  brief:  according  to  Theophrastus  all  laws  of  conversion  for 
problematic  sentences  are  exactly  analogous  to  those  for  assertoric 
sentences;  and  the  'reason'  of  which  Alexander  speaks,  can  only  be, 
so  it  would  seem,  that  the  modal  doctrine  of  Theophrastus  is  based 
on  one-sided  possibility,  while  Aristotle's  is  based  on  two-sided. 

Similarly  the  second  fundamental  thesis  of  the  Aristotelian  system 
is  also  rejected:  for  Theophrastus  the  functor  of  modality  must  be 
thought  of  as  determining  the  whole  sentence,  not  just  one  or  both 
of  its  arguments,  i.e.  it  must  be  thought  of  as  standing  at  the 
beginning  of  the  sentence  (cf.  commentary  on  15.13). 

17.15  But  his  companions  who  are  with  Theophrastus  and 
Eudemus,  deny  this,  and  say  that  all  formulae  consisting  of  a 
necessary  and  an  assertoric  premiss,  so  constituted  as  to  be 
suitable  for  syllogistic  inference,  yield  an  assertoric  conclusion. 
They  take  that  from  the  (principle  according  to  which)  in  all 
(syllogistic)  combinations  the  conclusion  is  similar  to  the  last 
and  weaker  premiss. 

17.16  But  Theophrastus,  (in  order  to  prove)  that  in  this 
combination  (auprXox?))  the  conclusion  yielded  is  not  necessary, 
proceeds  thus:  Tor  if  B  necessarily  (belongs)  to  C  and  A  does 
not  necessarily  belong  to  B,  if  one  disjoins  the  not  necessary, 
evidently,  as  B  is  disjoined  (from  A),  C  too  will  be  disjoined 
from  A :  hence  does  not  necessarily  belong  to  it  in  virtue  of 
the  premisses. 

17.17  They  prove  that  this  is  so  by  material  means  (=  by 
substitutions)  also.  For  they  take  a  necessary  universal 
affirmative    or   negative    as    major    (premiss),    an    assertoric 



universal  affirmative  as  minor,  and  show  that  these  yield  an 
assertoric  conclusion.  Suppose  that  animal  (belongs;  to  all 
man  necessarily,  but  man  belongs  (simply)  to  all  in  motion  : 
(then)  animal  will  not  necessarily  belong  to  all  in  motion. 

Now  the  basis  for  Aristotle's  permitting  the  drawing  of  a  necessary 
conclusion  from  one  necessary  and  one  assertoric  premiss,  was 
precisely  his  idea  of  the  structure  of  modal  sentences.  Theophrastue 
certainly  does  not  reject  this  idea  in  his  extant  fragments,  and  per- 
haps was  not  fully  aware  of  it.  But  in  any  case  all  that  we  have  of  his 
modal  logic  gives  evidence  of  a  system  presupposing  the  rejection 
of  the  Aristotelian  structure  of  modal  sentences. 


We  have  no  text  of  Theophrastus  that  contains  anything  of  his 
theory  of  hypothetical  propositions.  He  seems  to  have  treated  of 
them, for  he  distinguished  the  meaning  of  zl  and  iizzi  (17.18).  Possibly 
too  it  was  he  who  introduced  the  terminology  for  these  propositions 
which  Galen  aseribes  to  the  'old  Peripatetics'  (17.19).  However,  we 
know  that  he  developed  hypothetical  syllogisms: 

17.20  He  (Aristotle)  says  that  many  syllogisms  are  formed 
on  hypotheses.  .  .  .  Theophrastus  mentions  them  in  his 
Analytics,  as  do  Eudemus  and  some  others  of  his  companions. 

According  to  Philoponus,  both  of  them  'and  also  the  Stoics' 
wrote  'many-lined'  treatises  about  these  syllogisms  (17.21).  In  fact, 
however,  the  treatment  of  only  one  kind  of  these  syllogisms  is 
expressly  attributed  to  Theophrastus,  that  namely  which  consists  of 
'thoroughly  (St'  oXcov)  hypothetical'  syllogisms. 

17.22  However,  the  thoroughly  hypotheticals  are  reduced  to 
the  three  figures  in  another  way,  as  Theophrastus  has  proved  in 
the  first  book  of  the  Prior  Analytics.  A  thoroughly  hypothe- 
tical syllogism  is  of  this  kind:  If  A,  then  B;  if  B,  then  C;  if 
therefore  A,  then  C.  In  these  the  conclusion  too  is  hypotheti- 
cal; e.g.  if  man  is,  animal  is;  if  animal  is,  then  substance  is: 
if  therefore  man  is,  substance  is.  Now  since  in  these  too  there 
must  be  a  middle  term  in  which  the  premisses  convene  (for 
otherwise  here  also  there  cannot  be  a  conclusive  link),  this 
middle  will  be  positioned  in  three  ways.  For  if  one  premiss 
ends  with  it  and  the  other  begins  with  it,  there  will  be  the 
first  figure ;  it  will  be  in  fact  as  though  it  was  predicated  of  one 
extreme,  subjected  to  the  other.  ...  In  this  way  of  linking  one 



can  take  also  the  converse  of  the  conclusion,  in  such  a  way  that 
(C)  is  not  the  consequent  (E7r6[xevov)  but  the  antecedent 
(y]You(ji£vov),  not  indeed  simply,  but  with  opposition,  since 
when  a  conclusion  'if  A,  then  C"  has  been  gained,  there  is 
gained  a  conclusion  'if  not  C,  then  not  A\ 

If  the  premisses  begin  differently  and  end  similarly,  the 
figure  will  be  the  second,  like  the  second  (in  the  system)  of 
categorical  (syllogisms).  .  .  .  e.g.  If  man,  then  animal;  if 
stone,  then  not  animal;  therefore  if  man,  then  not  stone.  .  .  . 

If  the  premisses  begin  similarly  and  end  differently,  the 
figure  will  be  like  the  third  .  .  .  e.g.  if  A,  then  B ;  if  not  A,  then 
C;  it  will  follow :  therefore,  if  not  jB,  then  C,  or,  if  not  67,  then  B. 

The  formulae  contained  in  this  text  are  presented  in  such  a  way 
that  from  them  alone  it  is  impossible  to  tell  whether  their  variables 
are  term-  or  propositional  variables.  However,  the  substitutions 
show  that  the  former  is  the  case.  Hence  we  have  no  reason  to  ascribe 
any  law  of  propositional  logic  to  Theophrastus.  Yet  it  is  most 
probable  that  in  developing  Aristotle's  hints  about  'syllogisms  from 
hypotheses'  he  prepared  the  way  for  the  Megarian-Stoic  doctrine. 

The  formulae  in  the  text  just  cited  are  worded  as  rules;  but  we 
do  not  know  whether  this  wording  is  due  to  Theophrastus  himself, 
or  to  Alexander  and  so  mediately  to  the  Stoics. 





18.01  Euclid  originated  from  Megara  on  the  Isthmus.  .  .  . 
He  occupied  himself  with  the  writings  of  Parmenides;  his 
pupils  and  successors  were  called  'Megarians',  also  'Eristics' 
and  later  'Dialecticians'. 

18.02  The  Milesian  Eubulides  also  belongs  among  the 
successors  of  Euclid;  he  solved  many  dialectical  subtleties, 
such  as  The  Liar. 

18.03  Eubulides  was  also  hostile  to  Aristotle  and  made 
many  objections  to  him.  Among  the  successors  belonged 
Alexinus  of  Elis,  a  most  contentious  man,  whence  he  gained 
the  name  'Elenxinus'  ('Refuter'). 

18.04  Among  (the  pupils)  of  Eubulides  was  Apollonius, 
surnamed  Cronus,  whose  pupil  Diodorus,  the  son  of  Ameinias 
of  Iasus,  was  also  called  Cronus  .  .  ..  He  too  was  a  dialecti- 
cian. .  .  .  During  his  stay  with  Ptolemy  Soter  he  was  challenged 
by  Stilpo  to  solve  some  dialectical  problems;  but  as  he  could 
not  do  this  immediately  .  .  .,  he  left  the  table,  wrote  a  treatise 
on  the  problems  propounded,  and  died  of  despondency. 

18.05  Stilpo,  from  the  Greek  Megara,  studied  under  some 
pupils  of  Euclid;  others  say  that  he  studied  under  Euclid 
himself,  and  also  under  Thrasymachus  of  Corinth,  the  friend 
of  Icthyas.  He  surpassed  the  rest  in  inventiveness  of  argument 
and  dialectical  art  to  such  an  extent  that  well-nigh  all  Greece 
had  their  eyes  on  him  and  was  fain  to  follow  the  Megarian  school. 

18.06  He  caught  in  his  net  Crates  and  very  many  more. 
Among  them  he  captured  Zeno  the  Phoenician. 

18.07  Zeno,  the  son  of  Mnaseas  or  Demeas,  was  born  at 
Citium,  a  small  Greek  town  on  the  island  of  Cyprus,  where 
Phoenicians  had  settled. 

18.08  He  was  ...  a  pupil  of  Crates;  some  say  that  he  also 
studied  under  Stilpo. 

18.09  He  was  assiduous  in  discussion  with  the  dialectician 
Philo  and  studied  with  him;  so  that  Philo  came  to  be  admired 
by  the  more  youthful  Zeno  no  less  than  his  master  Diodorus. 



18.10  He  also  spent  some  time  under  Diodorus,  .  .  .  studying 
hard  at  dialectics. 

18.11  Kleanthes  the  son  of  Phanias  was  born  at  Assus  .  .  . 
joined  Zeno  .  .  .  and  remained  true  to  his  teaching. 

18.12  Chrysippus  the  son  of  Apollonius  from  Soli  or 
Tarsus  .  .  .,  was  a  pupil  of  Cleanthes. 

18.13  He  became  so  famous  as  a  dialectician,  that  it  was 
generally  said  that  if  the  gods  were  to  use  dialectic,  it  would  be 
none  other  than  that  of  Chrysippus. 

It  was  necessary  to  cite  these  extracts  from  the  Lives  and  Opinions 
of  Famous  Philosophers  of  Diogenes  Laertius,  in  order  to  counter  a 
widespread  error  to  the  effect  that  there  was  a  Stoic,  but  no  Megarian 
logic.  From  the  passages  quoted  it  appears  unmistakably  that  (a) 
the  Megarian  school  antedated  the  Stoic,  (b)  the  founders  of  the 
Stoa,  Zeno  and  Chrysippus,  learned  their  logic  from  the  Megarians, 
Diodorus,  Stilpo  and  Philo.  And  again  (c)  we  know  at  least  three 
Megarian  thinkers  of  importance  in  the  history  of  logic  -  Eubulides, 
Diodorus,  and  Philo  -  while  only  one  can  be  named  from  the  Stoa, 
viz.  Chrysippus  who  can  lay  claim  to  practically  no  basically  original 
doctrine,  whereas  each  of  the  three  Megarians  conceived  a  definitely 
original  idea. 

Admittedly  the  Megarian  school  seems  to  have  died  out  by  the 
close  of  the  3rd  century  b.c,  whereas  the  Stoa  continued  to  flourish. 
Also  the  adherents  of  the  latter  disseminated  logic  in  many  excellent 
handbooks  with  the  result  that  people,  as  in  Galen's  time,  spoke  only 
of  Stoic  logic.  The  least  that  can  justly  be  required  of  us  is  to  speak 
of  a  Megarian-Stoic  logic.  Possibly  the  basic  ideas  should  be  attri- 
buted to  the  Megarians,  their  technical  elaboration  to  the  Stoics, 
but  this  is  mere  conjecture. 

The  names  and  doctrinal  influences  recorded  by  Diogenes  can  be 
conveniently  summarized  in  the  following  table : 



Euclid  of  Megara,  pupil  of  Socrates, 

founder  of  the  Megarian  or  'dialectical'  school 

(ca.  400  b.c.) 

Alexinus  of  Elis 
called  'Elenxinus' 

of  Miletus 
of  The  Liar 



|  Thrasymachus 
friend  of 



Cronus  of  Iasus 
307  b.c.   — 

Philo  of  Megara 

Stilpo  of 


(ca.  320  b.c.) 


Zeno  of  Citium 
founder  of 
the  Stoa 
ca.  300  b.c. 

Cleanthes  of  Assus 

Chrysippus  of  Soli 
'second  founder  of 
the  Stoa' 

281/78-208/05(7)  b.c. 


The  conditions  for  investigation  of  the  Megarian-Stoic  logic 
are  much  less  favourable  than  those  for  that  of  the  logic  of  Aristotle 
or  even  Theophrastus.  We  have  the  essential  works  of  Aristotle 
entire,  and  in  the  case  of  Theophrastus  are  in  possession  at  least  of 
fragments  quoted  by  competent  experts  who  are  not  absolutely 
hostile  to  the  author  they  cite.  But  for  Megarian-Stoic  teaching  we 
have  to  rely  essentially  on  the  refutations  of  Sextus  Empiricus,  an 
inveterate  opponent.  As  B.  Mates  rightly  says,  it  is  as  though  we 
had  to  rely  for  a  knowledge  of  R.  Carnap's  logic  only  on  existen- 
tialist accounts  of  it.  Fortunately  Sextus,  though  no  friend  to  the 
Stoics,  was  (in  contrast  to  most  existentialists)  well  acquainted  with 
formal  logic,  which  he  opposed  from  his  sceptical  point  of  view. 
We  can  moreover  control  at  least  some  of  his  reports  by  means  of 
other  texts. 

But  still  we  have  nothing  but  fragments.  We  can  hardly  doubt 
that  the  material  to  hand  suffers  from  many  gaps:  for  instance 
term-logic  is  almost  completely  missing,  and  it  seems  hardly  likely 
that  it  was  wholly  unconsidered  in  the  Stoa. 



Another  problem  concerns  the  interpretation.  Even  in  antiquity 
Stoic  texts  were  often  'aristotelized',  propositional  variables  taken 
for  term-variables  etc.  A  similar  defect  characterizes  all  modern 
historians  of  logic,  Prantl  most  of  all,  who  completely  mistook  the 
significance  of  this  logic.  Peirce  was  the  first  to  see  that  it  was  a 
propositional  logic,  and  J.  Lukasiewicz  did  a  lasting  service  in  giving 
the  correct  interpretation.  Now  there  is  available  a  scientific 
monograph  -  a  rarety  in  history  of  logic  -  by  B.  Mates.  So  in  the 
present  state  of  research  it  can  be  stated  with  some  certainty  that 
we  are  again  in  a  position  to  understand  this  extremely  interesting 


In  reading  the  Megarian-Stoic  fragments  one's  first  impression 
is  that  here  is  something  different  from  Aristotelian  logic:  termino- 
logy, laws,  the  very  range  of  problems,  all  are  different.  In  addition 
we  are  confronted  with  a  new  technique  of  logic.  The  most  striking 
differences  are  that  the  Megarian-Stoic  logic  is  firstly  not  a  logic  of 
terms  but  of  propositions,  and  secondly  that  it  consists  exclusively 
of  rules,  not  of  laws  -  as  does  the  Prior  Analytics.  The  question  at 
once  arises,  what  was  the  origin  of  this  logic. 

The  answer  is  complex.  First  of  all  one  cannot  doubt  that  the 
Megarians  and  Stoics,  who  as  we  have  seen  (cf.  18.03)  found  an  only 
too  frequent  delight  in  refutation,  had  a  tendency  to  do  everything 
differently  from  Aristotle.  Thus  for  example  they  introduce  quite 
new  expressions  even  where  Aristotle  has  developed  an  excellent 

Yet  it  should  not  be  said  that  their  logical  thought  could  have 
developed  uninfluenced  by  Aristotle.  On  the  contrary,  they  appear  to 
have  developed  just  those  ideas  which  are  last  to  appear  in  the 
Organon.  We  find,  for  instance,  a  more  exact  formulation  of  the 
rules  which  Aristotle  used  in  axiomatizing  the  syllogistic,  and  him- 
self partially  formulated.  Nor  can  it  be  denied  that  they  developed 
his  theory  of  'syllogisms  from  hypotheses',  chiefly  on  the  basis  of 
the  preparatory  work  of  Theophrastus.  And  generally  speaking 
they  everywhere  show  traces  of  the  same  spirit  as  Aristotle's,  only 
in  a  much  sharper  form,  that  spirit  being  the  spirit  of  formalized 

And  that  is  not  yet  all.  In  many  of  his  non-analytical  formulae 
Aristotle  depends  directly  on  pre-Platonic  and  Platonic  discussions, 
and  this  dependence  is  still  greater  in  the  case  of  the  Megarian- 
Stoic  thinkers.  It  often  happens  that  they  transmute  these  discus- 
sions from  the  language  of  term-logic  into  that  of  propositional 
logic,  and  one  can  understand  how  they,  rather  than  Aristotle, 
came  to  do  this  on  such  a  scale.  Aristotle  always  remained  at  heart  a 



pupil  of  Plato's,  looking  for  essences,  and  accordingly  asking 
himself  the  question:  'Does  A  belong  to  B?'  But  the  Megarians  start 

from  the  pre-Platonic  question:  'How  can  the  statement  p  be 
refuted?'  Alexinus  was  called  'Hefuter',  and  all  these  thinkers 
continued  to  be  fundamentally  refuters  in  their  logic.  Which  means 
that  their  basic  problems  were  concerned  with  complete  propositions, 
whereas  Aristotle  had  his  attention  fixed  on  terms.  The  thorough 
empiricism  too,  to  which  the  Stoics  gave  their  allegiance,  contributed 
to  this  difference. 

As  to  details,  propositional  logic  originated  with  the  Megarians 
and  Stoics,  the  second  great  contribution  made  by  the  Greeks  to 
logic,  and  just  what  was  almost  entirely  missing  from  Aristotelian 
logic.  Then,  as  already  stated,  they  understood  formal  treatment  in 
a  formalistic  way,  and  laid  the  foundations  of  an  exact  semantics 
and  syntax.  Misunderstood  for  centuries,  this  logic  deserves  recogni- 
tion as  a  very  great  achievement  of  thought. 

Unfortunately  no  means  is  available  for  us  to  pursue  the  historical 
development  of  Megarian-Stoic  investigations;  we  can  only  consider 
what  we  find  at  the  end  of  this  development,  which  seems  to  have 
already  come  with  Chrysippus.  Within  a  hundred  and  fifty  years 
Greek  logic  rose  with  unbelievable  speed  to  the  very  heights  of 
formalism.  We  now  have  to  view  these  heights  as  already  attained. 
Our  presentation  cannot  be  historical;  it  can  only  proceed  systemati- 



19.01  They  (the  Stoics)  say  there  is  a  threefold  division  of 
philosophical  speech:  one  (part)  is  the  physical,  another  the 
ethical,  the  third  the  logical. 

19.02  They  compare  philosophy  to  an  animal,  the  logical 
part  corresponding  to  the  bones  and  sinews,  the  ethical  to  the 
fleshy  parts,  the  physical  to  the  soul.  Or  again  to  an  egg,  the 
logical  (part)  being  the  outside  (=the  shell).  ...  Or  again  to  a 
fertile  field.  The  fence  then  corresponds  to  the  logical. 

19.03  According  to  some  the  logical  part  is  divided  into 
two  sciences,  rhetoric  and  dialectic.  .  .  .  They  explain  rhetoric 
as  the  science  of  speaking  well  .  .  .  and  dialectic  as  the  science 
of  right  discussion  in  speech,  by  question  and  answer.  Hence  the 
following  definition:  it  is  the  science  of  the  true,  the  false,  and 
of  what  is  neither  of  the  two. 



That  of  course  does  not  mean  that  the  Stoics  knew  of  a  three- 
valued  logic  (cf .  49.08) ;  they  refer  only  to  sentences  (which  are  true 
or  false)  and  their  parts  (which  are  neither).  -  The  text  cited  expres- 
ses the  attitude  of  the  Stoics  to  the  problem  of  the  place  of  logic 
among  the  sciences :  for  them  it  is  quite  unmistakably  a  part  of  the 
system.  What  more  is  said  seems  to  concern  a  methodology  of  discus- 
sion rather  in  the  manner  of  the  Aristotelian  Topics  (11.01).  But  as 
we  know  from  other  fragments,  it  is  only  a  consequence  of  the  Stoic 
doctrine  of  the  principal  subject-matter  of  logic  which  consists  in 
lecia  (Xextoc).  This  important  notion  requires  immediate  clarifica- 


19.04  The  Stoics  say  that  these  three  are  connected:  the 
significate  (<yyj[jiaiv6[ji£vov),  the  sign  (<n)(Jiaivov)  and  the  thing 
(tuyxocvov).  The  sign  is  the  sound  itself,  e.g.  the  (sound)  'Dion', 
the  significate  is  the  entity  manifested  by  (this  sign)  and  which 
we  apprehend  as  co-existing  with  our  thought,  (but)  which 
foreigners  do  not  comprehend,  although  they  hear  the  sound; 
the  thing  is  the  external  existent,  e.g.  Dion  himself.  Of  these, 
two  are  bodies,  viz.  the  sound  and  the  thing,  and  one  imma- 
terial, viz.  the  entity  signified,  the  lecton,  which  (further)  is 
true  or  false. 

19.05  They  say  that  the  lecton  is  what  subsists  according 
to  a  rational  presentation  (xara  cpavTacuav  XoyixYjv). 

19.06  Some,  and  above  all  those  of  the  Stoa,  think  that 
truth  is  distinguished  in  three  ways  from  what  is  true,  .  .  . 
truth  is  a  body,  but  what  is  true  is  immaterial;  and  this  is 
shown,  they  say,  by  the  fact  that  what  is  true  is  a  proposition 
(a££<o[i.a),  while  a  proposition  is  a  lecton,  and  lecia  are  imma- 

We  have  refrained  from  translating  the  Greek  expression  Xsxtov 
which  derives  from  Xsyeiv  and  literally  means  'what  is  said',  i.e. 
what  one  means  when  one  speaks  meaningfully.  The  text  last  cited, 
about  truth  and  what  is  true,  is  to  be  specially  noted.  The  former  is 
something  psychic,  and  for  the  Stoics  all  such,  every  thought  in 
particular,  is  material.  But  the  lecton  is  not  a  quality  of  the  mind,  or 
in  scholastic  terminology  a  conceptus  subjectivus.  To  use  Frege's 
language  it  is  the  sense  (Sinn)  of  an  expression,  scholastically  the 
conceptus  objectivus,  what  is  objectively  meant.  In  the  (pseudo-) 
Aristotelian  Categories  there  is  a  passage  (10.29)  about  the  Xoyo?  tou 
7rpay(jLaTO^,  which  corresponds  to  the  Stoic  lecton.  Only  in  the  Stoa 



the  leclon  has  become  the  chief  subject-matter  of  logic  and  indeed 

the  unique  subject  of  formal  logic.  That  certainly  jettisons  the 
Aristotelian  neutrality  of  logic  arid  supposes  a  definite  philosophical 
standpoint.  But  the  original  philosophical  intuition  involved  is  to 
be  the  more  noticed  in  that  very  many  philosophers  and  logicians,  up 
to  the  most  recent  times,  have  confused  the  leclon  with  psychic 
images  and  occurrences  (cf.  26.07,  36.08).  That  the  Stoic  logic  is  a 
science  of  lecta  is  made  plain  by  their  division  : 

19.07  Dialectic  is  divided,  they  say,  into  the  topic  of 
signiflcates  and  (the  topic)  of  the  sound.  That  of  significates 
is  divided  into  the  topic  of  conceptions,  and  that  of  the  lecta 
which  co-exist  with  them:  propositions,  independents  (lecta), 
predicates,  and  so  on  .  . .,  arguments  and  moods  and  syllogisms, 
and  fallacies  other  than  those  arising  from  the  sound  and  the 
things.  ...  A  topic  proper  to  dialectic  is  also  that  already 
mentioned  about  the  sound  itself. 


19.08  The  elements  of  speech  are  the  twenty-four  letters. 
But  'letter'  can  have  three  meanings:  the  letter  (itself),  the 
(written)  sign  (^apaxr/jp)  of  the  letter,  and  its  name,  e.g. 
'alpha'.  .  .  .  Utterance  (<pcoWj)  is  distinguished  from  locution 
(Xe£is)  in  that  a  mere  sound  is  utterance,  but  only  articulated 
sound  is  locution.  Locution  is  distinguished  from  speech  in 
that  speech  is  always  meaningful,  while  what  has  no  meaning 
can  be  locution,  e.g.  'blityri'  -  which  is  not  speech. 

19.09  There  are  five  parts  of  speech,  as  Diogenes,  in  his 
(treatise)  On  Utterance,  and  Ghrysippus  say:  proper  names 
(ovofi-a),  general  names  (7rpo<77)Yopia),  verbs  (pjfxa),  connections 
(ouvSsctu,o?),  articles.  .  .  . 

19.10  A  general  name  is  according  to  Diogenes  a  part  of 
speech  which  signifies  a  common  quality,  e.g.  'man',  'horse'. 
But  a  proper  name  is  a  part  of  speech  which  manifests  a 
quality  proper  to  one,  e.g.  'Diogenes',  'Socrates'.  A  verb  is 
according  to  Diogenes  a  part  of  speech,  which  signifies  an 
incomposite  predicate  (xaTYjyopYjjjia),  or  as  others  (define  it),  an 
indeclinable  part  of  speech  which  signifies  something  co- 
ordinated with  one  or  more,  eg.  'I  write',  'I  speak'.  A  con- 
junction is  an  indeclinable  part  of  speech  which  connects  its 



Accounts  of  the  division  of  lecla  contradict  one  another,  and  are 
obscure.  The  following  scheme  composed  by  B.  Mates  (19.11)  may 
best  correspond  to  the  original  Stoic  teaching: 

What  is  said 


incomplete  complete 

zKknzic,  auTOTsXli; 

predicate  subject  proposition  others 

xaT7]YOp7](jLa  titcoctk;  a£ia)(j,a  7ttK7[ia 

But  the  division  of  propositions  is  clearly  and  fully  transmitted. 

19.12  A  proposition  is  what  is  true  or  false,  or  a  complete 
entity  (Trpayfxa)  assertoric  by  itself,  e.g.  'It  is  day*,  'Dion  walks 
about'.  It  is  called  'axiom'  (a£ico[xa)  from  being  approved 
(a£iouc7&ai)  or  disapproved.  For  he  who  says  'it  rs  day'  seems  to 
admit  that  it  is  day;  and  when  it  is  day,  the  foregoing  axiom  is 
true ;  but  when  it  is  not  (day),  false.  Different  from  one  another 
are  axiom,  question,  inquiry,  command,  oath,  wish,  exhorta- 
tion, address,  entity  similar  to  an  axiom. 

19.13  Of  axioms,  some  are  simple,  some  not  simple,  as  is 
said  in  the  schools  of  Chrysippus,  Archedemus,  Athenodorus, 
Antipater  and  Crinis.  Simple  are  those  which  consist  of  an 
axiom  not  repeated  (fjrf)  Swccpopou^evou),  e.g.  'it  is  day'.  Ones  not 
simple  are  those  consisting  of  a  repeated  axiom  or  of  more 
than  one  axiom.  An  example  of  the  former  is:  'if  it  is  day,  it  is 
day' ;  of  the  latter :  'if  it  is  day,  it  is  light' . 

19.14  Of  simple  axioms  some  are  definite  (cbpLcjjjiva),  others 
indefinite,  others  again  intermediate  (fxicja).  Definite  are  those 
which  are  referentially  expressed,  e.g.  'this  man  walks  about', 
'this  man  sits' :  (for  they  refer  to  an  individual  man).  Indefinite 
are  those  in  which  an  indefinite  particle  holds  the  chief  place 
(xupLEuei),  e.g.  'someone  sits'.  Intermediate  are  those  such  as: 
'  a  man  sits'  or  'Socrates  walks  about'.  .  .  . 

19.15  Among  axioms  not  simple  is  the  compound  (cjuvyj^fiivov 
=  conditional),  as  Chrysippus  in  the  Dialectic  and  Diogenes  in 
the  Dialectic  AH  say,  which  is  compounded  by  means  of  the 
implicative  connective  'if ;  this  connective  tells  one  that  the 
second  follows  from  the  first,  e.g.  'if  it  is  day,  it  is  light'.  An 
inferential  axiom  (7rapaauvy)[i.(xevov)  is,  as  Crinis  says  in  the 
Dialectic  Art,  one  which  begins  and  ends  with  an  axiom  and  is 



compounded  7capamwJ7CTai)  by  means  of  the  connective  'since1 
(inei),  e.g.  'since  it  is  day,  it  is  light'.  This  connective  tells  one 
that  the  second  follows  from  the  first  and  that  the  first  is  the 
case.  Conjunctive  (au[jwus7cXeYjjivov)  is  the  axiom  compounded  by 
means  of  a  conjunctive  connective,  e.g.  'it  is  day  and  it  is 
light'.  Disjunctive  (Sis^euytiivov)  is  the  axiom  compounded  by 
means  of  the  separative  connective  'or',  e.g.  'it  is  day  or  it  is 
night'.  This  connective  tells  one  that  one  of  the  axioms  is 
false.  Causal  (amto<k<;)  is  the  axiom  compounded  by  means 
of  the  connective  'because',  e.g.  'because  it  is  day,  it  is  light'. 
For  it  is  here  to  be  understood  that  the  first  is  the  cause  of  the 
second.  An  axiom  showing  what  is  rather  the  case  is  one  com- 
pounded by  means  of  the  connective  'rather  than'  which 
shows  this  and  stands  in  the  middle  of  the  axiom,  e.g.  'it  is 
night  rather  than  day'. 

Note  in  these  texts  that  lecla,  not  words  or  psychic  events  are  the 
subject-matter  throughout.  Hence  most  translations  (those  e.g.  of 
Apelt,  19.16,  and  Hicks)  are  misleading,  since  they  talk  of  connective 
'words'  and  'judgements'. 


19.17  The  common  genus  'what  is'  has  nothing  over  it.  It 
is  the  beginning  of  things  and  everything  is  inferior  to  it.  The 
Stoics  wanted  to  put  another,  still  more  principal  genus  above 

19.18  To  some  Stoics  'what'  seems  to  be  the  prime  genus; 
and  I  will  say  why.  In  nature,  they  say,  some  things  exist, 
others  do  not.  Even  those  which  do  not  exist  are  contained  in 
nature,  those  which  occur  in  the  soul,  like  centaurs,  giants 
and  anything  else  which  acquires  an  image  when  falsely 
framed  in  thought,  though  having  no  substance. 

So  according  to  these  Stoics  there  is  a  summum  genus.  This  is  a 
regression  in  comparison  with  Aristotle's  subtle  anticipation  of  a 
theory  of  types  (11.16). 

19.19  But  the  Stoics  think  that  the  prime  genera  are  more 
limited  in  number  (than  the  Aristotelian).  .  .  .  For  they 
introduce  a  fourfold  division  into  subjects  (u7rox£iji.sva),  qualia 
(tcoloc),  things  that  are  in  a  determinate  way  (ttco;  £'x0VTa)>  and 
things  that  are  somehow  related  to  something  (r.pbc,  ti  -co; 



These  four  categories  are  not  to  be  understood  as  supreme  genera 
(under  the  'what'):  That  is,  it  is  not  the  case  that  one  being  is  a 
subject,  another  a  relation,  but  all  the  categories  belong  to  every 
being,  and  every  category  presupposes  the  preceding  ones  (19.20). 
This  doctrine  has  no  great  significance  for  logic. 


Apart  from  the  distinction  already  mentioned  between  what  is 
true  and  truth  (19.06),  the  Stoics  seem  to  have  used  the  word  'true' 
in  at  least  five  senses.  On  this  Sextus  says : 

19.21  Some  of  them  have  located  the  true  and  the  false  in 
the  significates  (  =leda),  others  in  the  sound,  others  again  in 
the  operation  of  the  mind. 

As  regards  the  truth  of  lecla,  a  further  threefold  distinction  can  be 
made  between: 

1.  truth  of  propositions. 

2.  truth  of  propositional  forms  (i.e.  what  it  is  that  sentential 
functions  refer  to).  That  the  Megarians  and  Stoics  attributed  truth 
and  falsity  to  such  propositional  forms  is  seen  in  their  teaching 
about  functors  (vide  infra). 

3.  truth  of  arguments  (vid.  21.07). 

Those  are  all  lecla,  but  Sextus  refers  to  two  further  kinds  of  truth: 

4.  truth  of  ideas  (19.22). 

5.  truth  of  sentences. 

According  to  all  our  information  the  first  kind  of  truth  was 
fundamental,  as  presupposed  in  all  the  others.  Thus  for  instance  the 
Stoics  defined  the  truth  of  propositional  forms  by  its  means,  with 
the  help  of  time-variables;  the  truth  of  arguments  in  terms  of  the 
truth  of  the  corresponding  conditional  propositions;  while  the  truth 
of  ideas  and  sentences  is  similarly  reducible  to  that  of  lecla,  accord- 
ing to  what  we  know  of  the  relation  between  them. 


Only  fragments  have  come  down  to  us  of  the  very  interesting 
Megarian  doctrine  of  modalities.  It  seems  to  be  an  attempt  to  reduce 
necessity  and  possibility  to  simple  existence  by  means  of  time- 
variables,  a  proceeding  wholly  consonant  with  the  empirical  stand- 
point of  these  thinkers.  We  give  only  the  two  most  important 
passages  on  the  subject: 

19.23  'Possible'  can  also  be  predicated  of  what  is  possible  in 
a  'Diodorean'  sense,  that  is  to  say  of  what  is  or  will  be.  For  he 



(Diodorus)  deemed  possible  only  what  either  is  or  will  be. 
Since  according  to  him  it  is  possible  that  I  am  in  Corinth,  if 
I  am  or  ever  shall  be  there;  and  if  I  were  not  going  to  be 
there,  it  would  not  be  possible.  And  it  is  possible  that  a  child 
should  be  a  grammarian,  if  he  will  ever  be  one.  To  prove  this, 
Diodorus  devised  the  master-argument  (xupiefoov).  Philo  took  a 
similar  view. 

19.24  The  (problem  of)  the  master-argument  seem-  prob- 
ably to  have  originated  from  the  following  considerations.  As 
the  following  three  (propositions)  are  incompatible:  (1) 
Whatever  is  true  about  the  past  is  necessary,  (2)  the  impos- 
sible does  not  follow  from  the  possible  which  neither  is  nor 
will  be  true  -  Diodorus,  comparing  this  incompatibility  with 
the  greater  plausibility  of  the  first  two,  inferred  that  nothing  is 
possible  which  neither  is  nor  will  be  true. 

Unfortunately  that  is  the  only  really  explicit  text  about  the 
celebrated  master-argument  of  Diodorus.  It  fails  to  enable  us  to 
survey  the  whole  problem,  because  we  do  not  know  why  the  three 
propositions  should  be  incompatible.  One  thing  seems  clear:  that 
possibility  was  defined  in  the  following  way : 

19.241  p  is  (now)  possible  if  and  only  if  p  is  now  true  or 
will  be  true  at  some  future  time. 

From  a  rather  vague  text  of  Boethius  (19.25)  we  further  learn 
that  the  definitions  of  the  other  possibility-functors  must  be  more 
or  less  as  follows : 

19.242  p  is  (now)  impossible  if  and  only  if  p  is  not  true  and  never 
will  be  true. 

19.243  p  is  (now)  necessary  if  and  only  if  p  is  true  and  always  will 
be  true. 

19.244  p  is  (now)  not  necessary  if  and  only  if  p  is  not  true  or  will 
not  be  true  at  some  future  time. 


To  the  credit  of  the  Megarian-Stoic  school  are  some  very  subtle 
researches  into  the  most  important  propositional  functors.  The 
thinkers  of  the  school  even  succeeded  in  stating  quite  correct 



20.01  Negative  are  said  to  be  only  those  propositions  to 
which  the  negative  particle  is  prefixed. 

This  text  shows  something  to  which  many  passages  bear  witness, 
that  the  Stoics  constructed  their  logic  not  merely  formally,  but  quite 
formalistically.  This  was  blamed  by  Apuleius  (20.02)  and  Galen 
(20.03),  who  said  that  the  Stoics  were  only  interested  in  linguistic 
form.  But  this  reproach  -  if  indeed  it  is  one  -  cannot  be  sustained 
in  view  of  what  we  know  of  the  subject-matter  of  Stoic  logic;  Stoic 
formalism  is  concerned  with  words  only  as  signs  of  lecta. 

20.04  Among  simple  axioms  are  the  negative  (darocpaTLxov), 
the  denying  (apv7)Ti.x6v),  the  privative  ((jTepyraxov).  .  .  .  An 
example  of  the  negative  is:  'it  is  not  day'.  A  species  of  this 
is  the  super-negative  (u7repoc7rocpaT[.x6v).  The  super-negative  is 
the  negation  of  the  negative,  e.g.  'not  -  it  is  not  day'.  This 
posits  'it  is  day'.  A  denying  (axiom)  is  one  which  consists  of 
a  negative  particle  and  a  predicate,  e.g.  'No-one  walks  about'. 
Privative  is  one  which  consists  of  a  privative  particle  and 
what  has  the  force  of  an  axiom,  e.g.  'this  man  is  unfriendly 
to  man'. 

The  extant  fragments  do  not  contain  a  table  of  truth-values  for 
negation,  but  the  text  cited  evidently  contains  the  law  of  double 
negation : 

20.041  not-not  p  if  and  only  if  p  (cf.  24.26). 


The  definition  of  implication  was  a  matter  much  debated  among 
the  Megarians  and  Stoics: 

20.05  All  dialecticians  say  that  a  connected  (proposition)  is 
sound  (uyti<;)  when  its  consequent  follows  from  (axoAoufrsi)  its 
antecedent  -  but  they  dispute  about  when  and  how  it  follows, 
and  propound  rival  criteria. 

Even  so  Callimachus,  librarian  at  Alexandria  in  the  2nd  century 
b.c,  said: 

20.06  The  very  crows  on  the  roofs  croak  about  what 
implications  are  sound. 



1.  Philonian  implication 

20.07  Philo  said  that  the  connected  (proposition)  is  true 
when  it  is  not  the  case  that  it  begins  with  the  true  and  ends 
with  the  false.  So  according  to  him  there  are  three  ways  in 
which  a  true  connected  (proposition)  is  obtained,  only  one  in 
which  a  false.  For  (1)  if  it  begins  with  true  and  ends  with  true, 
it  is  true,  e.g.  'if  it  is  day,  it  is  light' ;  (2)  when  it  begins  with  false 
and  ends  with  false,  it  is  true,  e.g.  'if  the  earth  flies,  the  earth 
has  wings';  (3)  similarly  too  that  which  begins  with  false  and 
ends  with  true,  e.g.  'if  the  earth  flies,  the  earth  exists'.  It  is 
false  only  when  beginning  with  true,  it  ends  with  false,  e.g. 
'if  it  is  day,  it  is  night' ;  since  when  it  is  day,  the  (proposition) 
'it  is  day'  is  true  -  which  was  the  antecedent ;  and  the  (proposi- 
tion) 'it  is  night'  is  false,  which  was  the  consequent. 

Here  some  terminological  explanations  are  required.  The  Stoics 
called  the  antecedent  "/jyouuxvov,  the  consequent  Xvjyov,  and  moreover 
had  the  corresponding  verbs:  ^yetTOu,  Xyjyei,  untranslatable  in  their 
technical  use.  Hence  we  have  simply  translated  these  words  accord- 
ing to  their  ordinary  sense,  by  'begins'  and  'ends'.  The  term  too  for 
the  sentences  themselves  (or  the  propositions  to  which  they  refer)  has 
been  translated  according  to  its  everyday  sense  by  'connected',  the 
word  'conditional'  having  been  avoided  because  apparently  the 
idea  of  condition  was  foreign  to  Megarian-Stoic  thought. 

As  to  the  content  of  the  passage,  it  gives  us  a  perfect  truth- 
matrix,  which  can  be  set  out  in  tabular  form  thus: 

















It  is,  as  we  can  see,  the  truth-value  matrix  for  material  impli- 
cation, ordered  otherwise  than  is  usual  nowadays  (41.12;  but  42.27). 
The  latter  therefore  deserves  to  be  called  'Philonian'. 

2.  Diodorean  implication 

20.08  Diodorus  says  that  the  connected  (proposition)  is 
true  when  it  begins  with  true  and  neither  could  nor  can  end 
with  false.  This  runs  counter  to  the  Philonian  position.  For  the 
connected  (proposition)  'if  it  is  day,  I  converse'  is  true 
according  to  Philo,  in  case  it  is  day  and  I  converse,  since  it 



begins  with  the  true  (proposition)  'it  is  day'  and  ends  with  the 
true  (proposition)  'I  converse'.  But  according  to  Diodorus  (it  is) 
false.  For  at  a  given  time  it  can  begin  with  the  true  (proposi- 
tion) 'it  is  day'  and  end  with  the  false  (proposition)  ' I  converse' , 
suppose  I  should  fall  silent  .  .  .  (and)  before  I  began  to  con- 
verse it  began  with  a  true  (proposition)  and  ended  with  the 
false  one  'I  converse'.  Further,  the  (proposition)  'if  it  is 
night,  I  converse'  is  true  according  to  Philo  in  case  it  is  day 
and  I  am  silent;  for  it  (then)  begins  with  false  and  ends  with 
false.  But  according  to  Diodorus  (it  is)  false;  for  it  can  begin 
with  true  and  end  with  false,  in  case  the  night  is  past  and 
I  am  not  conversing.  And  also  the  (proposition)  'if  it  is  night, 
it  is  day'  is  according  to  Philo  true  in  case  it  is  day,  because, 
while  it  begins  with  the  false  (proposition)  'it  is  night',  it  ends 
with  the  true  (proposition)  'it  is  day'.  But  according  to  Dio- 
dorus it  is  false  because,  while  it  can  begin  -  when  night  is 
come  -  with  the  true  (proposition)  'it  is  night',  it  can  end  with 
the  false  (proposition)  'it  is  day'. 

So  we  can  fix  Diodorean  implication  by  the  following  definition : 

20.081  If  p,  then  q,  if  and  only  if,  for  every  time  I  it  is 
not  the  case  that  p  is  true  at  t  and  q  is  false  at  t. 

3.  'Connexive''  implication 

20.09  (According  to  Diodorus)  this  (proposition)  is  true: 
'if  there  are  no  atomic  elements  of  things,  then  there  are 
atomic  elements  of  things'  .  .  .  but  those  who  introduce  con- 
nection ((TuvapTTjcjLv)  say  that  the  connected  (proposition)  is 
sound  when  the  contradictory  (avTt,x£i[i.s:vov)  of  its  consequent 
is  incompatible  ((xdcx^Tai)  with  its  antecedent.  So  according  to 
them  the  aforesaid  connected  (propositions)  (20.07)  are  bad 
({AoxOvjpa),  but  the  following  is  true  (<xXyfi£q) :  'if  it  is  day,  it  is 

20.10  A  connected  (proposition)  is  true  in  which  the  opposite 
of  the  consequent  is  incompatible  with  the  antecedent,  e.g. 
'if  it  is  day,  it  is  light'.  This  is  true,  since  'it  is  not  light',  the 
opposite  of  the  consequent,  is  incompatible  with  'it  is  day'. 
A  connected  (proposition)  is  false  in  which  the  opposite  of  the 
consequent  is  not  incompatible  with  the  antecedent,  e.g.  'if  it 
is  day,  Dion  walks  about';  for  'Dion  is  not  walking  about'  is 
not  incompatible  with  'it  is  day'. 



This  definition  is  often  ascribed  to  Chrysippus  (20.11),  but  that 
it  originated  with  him  may  be  doubted  (20.12).  It  is  not  clear  how- 
it  is  to  be  understood.  Perhaps  we  have  here  an  ancient  form  of 
strict  implication  (49.04  31.13). 

4.  '  Inclusive'  implication 

20.13  Those  who  judge  (implication)  by  what  is  implicit 
(eu-cpaaei  xptvovT£<;)  say  that  the  connected  (proposition)  is 
true  when  its  consequent  is  potentially  (Suvdqiei)  contained  in 
the  antecedent.  According  to  them  the  (proposition)  'if  it  is 
day,  it  is  day'  and  every  repetitive  connected  (proposition)  is 
probably  false,  since  nothing  can  be  contained  in  itself. 

This  definition  too  is  not  now  fully  intelligible.  It  seems  to  concern 
a  relation  of  subordination  something  like  that  which  holds  between 
a  statement  about  all  elements  of  a  class  and  one  about  the  elements 
of  one  of  its  sub-classes.  No  further  reference  to  this  definition  is  to 
be  found  in  our  sources;  perhaps  it  was  only  adopted  by  isolated 
logicians  of  the  school. 


We  know  much  less  about  disjunction  than  about  implication. 
Apparently  it  formed  the  subject  of  the  same  sort  of  dispute  that 
there  was  about  the  definition  of  implication.  But  our  texts  are  few 
and  obscure.  It  is  only  certain  that  two  kinds  of  disjunction  were 
recognized :  the  complete  (exclusive)  and  the  incomplete  (not  exclu- 
sive), of  which  the  first  is  well  exemplified. 

1.  Complete  disjunction 

20.14  The  disjunctive  (proposition)  consists  of  (contra- 
dictorily) opposed  (propositions),  e.g.  of  those  to  the  effect 
that  there  are  proofs  and  that  there  are  not  proofs.  .  .  .  For  as 
every  disjunctive  is  true  if  (and  only  if)  it  contains  a  true 
(proposition)  and  since  one  of  (two  contradictorily)  opposed 
(propositions)  is  evidently  always  true,  it  must  certainly  be 
said  that  the  (proposition)  so  formed  is  true. 

20.15  There  is  also  another  (proposition)  which  the  Greeks 
call  Sis^euyuivov  a£icou.a  and  we  call  disjunctum.  This  is  of  the 
kind :  'pleasure  is  either  good  or  bad,  or  neither  good  nor  bad'. 
Now  all  (propositions)  which  are  disjoined  (disjuncta)  (within 
one  such  proposition)  are  mutually  incompatible,  and  their 
opposites,  which  the  Greeks  call  avn.xsiu.sva  must  also  be 
mutually  opposed  (contraria).  Of  all  (propositions)  which  are 
disjoined,  one  will  be  true,  the  others  false.  But  when  none  of 



them  at  all  are  true,  or  all,  or  more  than  one  are  true,  or  when 
the  disjoined  (propositions)  are  not  incompatible,  or  when 
their  opposites  are  not  mutually  opposed,  then  the  disjunctive 
(proposition)  will  be  false.  They  call  it  TOxpocSis^euyfiivov. 

20.16  The  true  disjunctive  (proposition)  tells  us  that  one  of 
its  propositions  is  true,  the  other  or  others  false  and  incompat- 

These  texts  offer  a  difficulty,  in  the  supposition  that  a  statement 
can  be  contradictory  to  more  than  one  other.  However,  the  practice 
of  the  school  concerning  the  disjunction  here  defined  is  clear:  in  the 
sense  envisaged  'p  or  q'  is  understood  as  the  negation  of  equivalence 
(vide  infra  22.07),  i.e.  in  such  a  way  that  just  one  of  the  two  argu- 
ments is  true  and  just  one  false. 

2.  Incomplete  disjunction 

The  surviving  information  about  this  is  very  vague.  The  best  is 
given  by  Galen,  but  raises  the  question  how  much  of  it  is  Megarian- 
Stoic  doctrine  and  how  much  Galen's  own  speculation: 

20.17  This  state  of  things  exhibits  a  complete  incompat- 
ibility (tiXeiav  paxyjv),  the  other  an  incomplete  (sXXottjv) 
according  to  which  we  say  for  example:  'if  Dion  is  at  Athens, 
Dion  is  not  at  the  Isthmus'.  For  this  is  characteristic  of  in- 
compatibility, that  incompatibles  cannot  both  be  the  case; 
but  they  differ  in  that  according  to  the  one  the  incompatibles 
can  neither  both  be  true  nor  both  false,  but  according  to  the 
other  this  last  may  occur.  If  then  only  inability  to  be  true 
together  characterizes  them,  the  incompatibility  is  incomplete, 
but  if  also  inability  to  be  false  together,  it  is  complete. 

20.18  There  is  no  reason  why  we  should  not  call  the  propos- 
ition involving  complete  incompatibility  'disjunctive'  and  that 
involving  incomplete  incompatibility  'quasi-disjunctive'.  .  .  . 
But  in  some  propositions  not  only  one,  but  more  or  all 
components  can  be  true,  and  one  must  be.  Some  call  such 
'sub-disjunctive'  (7cap<x8is£euY(Aeva) ;  these  contain  only  one 
true  (proposition)  among  those  disjoined,  independently  of 
whether  they  are  composed  of  two  or  more  simple  propositions. 

Evidently  this  is  a  matter  of  two  different  kinds  of  disjunctive 
propositions,  and  so  of  disjunction.  The  first  is  called  'quasi-disjunc- 
tion'  and  seems  to  be  equivalent  to  the  denial  of  conjunction: 

20.181  p  or  q  if  and  only  if,  not:  p  and  q. 


Then  the  intended  functor  would  be  that  of  Shelter  (43.43). 
The  second  kind  is  called  'sub-disjunctive'  and  could  be  defined 
by  the  following  equivalence: 

20.182  p  or  (also)  q  if  and  only  if:  if  not  p,  then  q. 

This  is  the  modern  functor  of  the  logical  sum  (cf.  14.1011.  . 

Neither  of  these  two  functors  was  used  by  the  Stoics  in  practice, 
at  least  so  far  as  we  can  ascertain  from  the  extant  sources. 


20.19  What  the  Greeks  call  (jufJwue7cXeYjjtivov  we  call  conjuncium 

or  copulatum.  It  is  as  follows :  'Publius  Scipio,  son  of  Paulus,  was  twice 
consul  and  had  a  triumph  and  was  censor  and  was  colleague  of 
Lucius  Mummius  in  the  censorship.'  In  every  conjunctive  the  whole 
is  said  to  be  false  if  one  (component)  is  false,  even  if  the  others  are 
true.  For  if  I  were  to  add  to  all  that  I  have  truly  said  about  that 
Scipio:  'and  overcame  Hannibal  in  Africa',  which  is  false,  then  the 
whole  conjunctive  which  includes  that  would  be  false:  because 
that  is  a  false  addition,  and  the  whole  is  stated  together. 


20.20  Syllogisms  which  have  hypothetical  premisses  are 
formed  by  transition  from  one  thing  to  another,  because  of 
consequence  (axoXou0La)  or  incompatibility,  each  of  which 
may  be  either  complete  or  incomplete. 

20.21  The  (exclusive)  disjunctive  premiss  (Siaiperwdj)  is 
equivalent  to  the  following:  'if  it  is  not  day,  it  is  night'. 

This  last  cited  text,  in  which  quite  certainly  complete  disjunction 
is  intended  (20.14  ff.)  can  only  be  understood  as  referring  to  'com- 
plete consequence'  (20.20)  -  and  then  we  have  equivalence.  In  this 
case  we  have  the  following  definition,  in  which  'or'  is  to  be  under- 
stood in  the  exclusive  sense: 

20.211   q  completely  follows  from  p  if  and  only  if,  not :  p  or  q. 

We  owe  the  discovery  of  these  facts  to  Stakelum  (20.22).  Boethius, 
probably  drawing  on  a  Stoic  source,  understands  'if  A  -  B'  in  just 
this  sense  20.23).  So  it  can  be  taken  as  likely  that  the  functor  of 
equivalence  was  known  to  the  Stoics  as  'complete  consequence'. 


We  also  have  definitions  of  the  inferential  proposition  (cf.  19.15  . 
This  consists  of  a  combination  of  conjunction  with  Diodorean 
(certainly   not   Philonian)   implication.    Other  kinds   of  compound 



propositions  are  the  causal  and  the  relative;  their  functors  are  not 
definable  by  truth-matrices.  Possibly  there  are  further  functors  of 
similar  nature. 



21.01  An  argument  (Xoyo^)  is  a  system  of  premisses  and 
conclusion.  Premisses  are  propositions  agreed  upon  for  the 
proof  of  the  conclusion,  the  conclusion  is  the  proposition 
proved  from  the  premisses.  E.g.  in  the  following  (argument): 
'if  it  is  day,  it  is  light;  it  is  day;  therefore  it  is  light',  'it  is 
light'  is  the  conclusion,  the  other  propositions  are  premisses. 

21.02  Some  arguments  are  conclusive  (cjuvocxtixol),  others 
not  conclusive.  They  are  conclusive  when  a  connected 
proposition,  beginning  with  the  conjunction  of  the  premisses 
of  the  argument  and  ending  with  the  conclusion,  is  true. 
E.g.  the  argument  mentioned  above  is  conclusive,  since  from 
the  conjunction  of  its  premisses  'if  it  is  day,  it  is  light'  and  'it  is 
day'  there  follows  'it  is  light'  in  this  connected  proposition: 
'if:  it  is  day  and  if  it  is  day,  it  is  light:  then  it  is  light.'*  Not 
conclusive  are  arguments  not  so  constructed. 

This  is  a  very  important  text,  showing  how  accurately  the  Stoic 
distinguished  between  a  conditional  proposition  and  implication 
on  the  one  hand,  and  an  argument  or  inferential  scheme  and  the 
consequence-relation  on  the  other.  For  an  argument  is  conclusive 
(cruvaxTLxo^)  when  the  corresponding  conditional  proposition  is  true 

The  Stoics  had  a  set  terminology  for  the  components  of  an  argu- 
ment. In  the  simplest  case  it  has  two  premisses,  X^jJiaTa  (in  the 
wider  sense);  the  first  is  also  called  XyjfjLfxa  (the  narrower  sense),  in 
contrast  to  the  second  which  is  called  izpoa'krityic,  21.04);  when  the 
first  premiss  is  connected,  it  is  also  called  Tpomxov  (21.05). 

21.06  Of  arguments,  some  are  not  conclusive  (aTuepavToi), 
others  conclusive  (nzpoLvrixoi) .  Not  conclusive  are  those  in 
which  the  contradictory   opposite   of  the  conclusion  is  not 

Reading  etnep  el  rjjjipa  sax£,  xal  ■yjjxepa  eari,  cpcot;  ecrdv.  This  reading  was 
called  a  'monstrosity'  by  Heintz,  whereas  it  is  evidently  the  only  correct  one 



incompatible  with  the  conjunction  of  the  premisses,  e.g.  such 
as:  'if  it  is  day,  it  is  light;  it  is  day;  therefore  Dion  walks 

It  seems  to  follow  that  the  conditional  sentence  corresponding 
to  an  argument  must  contain  the  functor  of  connexive  implication 
for  an  argument  to  be  conclusive  (cf.  20.09 f.  and  the  commentary). 

21.07  Of  conclusive  arguments  some  are  true,  others  false. 
They  are  true  when  besides  the  connected  proposition,  which 
consists  of  the  conjunction  of  the  premisses  and  the  conclusion, 
being  true,  the  conjunction  of  the  premisses  is  also*  true,  i.e. 
that  which  forms  the  antecedent  in  the  connected  proposition. 

Again  a  text  of  the  utmost  importance,  expressing  a  clear  distinc- 
tion between  formal  validity  and  truth.  This  distinction  was  admit- 
tedly known  to  Aristotle  (10.05 f.),  but  this  is  the  first  explicit 
accurate  formulation. 

21.09  Of  true  arguments  some  are  demonstrative  (<x7co&eix- 
tlxol)  others  not  demonstrative.  Demonstrative  are  those  con- 
cluding to  the  not  evident  from  the  evident,  not  demonstrative 
are  those  not  of  that  kind.  E.g.  the  argument:  'if  it  is  day,  it  is 
light;  it  is  day;  therefore  it  is  light'  is  not  demonstrative,  for 
that  it  is  light  (which  is  evident)  is  its  conclusion.  On  the  other 
hand,  this  is  demonstrative:  'if  the  sweat  flows  through  the 
surface,  there  are  intelligible  (voyjtol)  pores;  the  sweat  flows 
through  the  surface;  therefore  there  are  intelligible  pores',  for 
it  has  a  non-evident  conclusion,  viz.  'therefore  there  are 
intelligible  pores'. 

'Intelligible'  here  means  'only  to  be  known  by  the  mind';  the 
pores  are  not  visible. 

The  division  of  arguments  comprised  in  this  last  series  of  texts  is 
logically  irrelevant,  but  of  great  methodological  interest.  It  can  be 
presented  thus: 



[    true                I 

(    conclusive         I                          { 

not  demonstrative 

arguments    I                                [    not  true 

(    not  conclusive 

*   Omitting  with  Mates  (21.08):  xal  to  G\j[iniptxo\±ot.. 



A  further  interesting  division  shows  how  accurately  the  Stoics 
distinguished  between  language  and  meta-language : 

21.10  Of  conclusive  arguments,  some  are  called  by  the  name 
of  the  genus,  'conclusive'  (rapavTLxoL),  others  are  called  'syl- 
logistic'. Syllogistic  are  those  which  are  either  indemonstrable 
(ava7r6<teixTOL)  or  are  reduced  to  the  indemonstrable  by  means 
of  one  or  more  rules  (tc5v  0£u.dcTcov),  e.g.  'if  Dion  walks  about, 
Dion  is  in  motion;  Dion  walks  about;  therefore  Dion  is  in 
motion'.  Conclusive  in  the  specific  sense  are  those  which  do  not 
conclude  syllogistically,  those  of  e.g.  the  following  kind :  'it  is 
false  that  it  is  day  and  it  is  night;  it  is  day;  therefore  it  is  not 
night'.  Non-syllogistic,  on  the  other  hand,  are  arguments 
which  appear  to  resemble  syllogistic  ones,  but  do  not  conclude, 
e.g.  'if  Dion  is  a  horse,  Dion  is  an  animal;  Dion  is  not  a  horse; 
therefore  Dion  is  not  an  animal'. 

21.11  .  .  .  but  the  moderns,  who  follow  the  linguistic 
expression,  not  what  it  stands  for,  .  .  .  say  that  if  the  expres- 
sion is  formulated  thus:  'if  A,  then  B;  A;  therefore  B,'  the 
argument  is  syllogistic,  but  'B  follows  on  A;  A;  therefore  B' 
is  not  syllogistic,  though  it  is  conclusive. 

21.12  .  .  .  The  kind  of  argument  which  is  called  'unmetho- 
dically concluding'  (au^OoSox;  7cepaivovTes)  is  e.g.  this:  'it  is 
day;  but  you  say  that  it  is  day;  therefore  you  say  true'. 

21.13  (Those  which  the  moderns  call  'unmethodically 
concluding'  .  .  .)  are  such  as  the  following:  'Dion  says  that  it  is 
day;  Dion  says  true;  therefore  it  is  day'. 

21.14  .  .  .  like  the  unmethodically  concluding  arguments 
among  the  Stoics.  When  e.g.  someone  says:  'the  first  (is) 
greater  than  the  second,  the  second  than  the  third,  therefore 
the  first  (is)  greater  than  the  third'. 


21.15  Those  arguments  too  which  they  call  'duplicated' 
(SioccpopoVevoi)  are  n°t  syllogistic,  e.g.  this:  'if  it  is  day,  it  is 
day;  therefore  it  is  day'. 

21.16  The  argument):  'if  it  is  day,  it  is  light;  it  is  day; 
therefore  it  is  day',  and  in  general  those  which  the  moderns 
call  'not  diversely  concluding'  (aSt^opo^  rcepaivovTe^) .  .  .  . 


M  E  G  A  R  I A  N  -  S  T  O  I C   LOGIC 

21.17  Antipater,  one  of  the  most  celebrated  men  of  the 
Stoic  sect,  used  to  say  that  arguments  with  a  single  premiss 
can  also  be  formed  (|jiovoXy)(IU,<xtol). 

21.18  From  one  premiss  there  results  no  (conclusive)  com- 
bination (colledio),  though  the  consequence  (conclusio)  'you 
see,  therefore  you  live'  seemed  complete  to  Antipater  the 
Stoic,  against  the  doctrine  of  all  (others)  -  for  it  is  complete 
(only)  in  the  following  way;  'if  you  see,  then  you  live;  you  see; 
therefore  you  live'. 

21.19  Such  an  argument  as  that  which  says:  'it  is  day;  not: 
it  is  not  day;  therefore  it  is  light'  has  potentially  a  single 

Further,  apparently  numberless,  divisions  of  arguments  are 
obscure  in  our  sources.  Diogenes  speaks  of  'possible,  impossible, 
necessary  and  not  necessary'  arguments  (21.20).  Sextus  has  a 
division  into  demonstrable  and  indemonstrable  arguments,  the 
last-named  being  either  simple  or  compound,  and  the  compound 
being  reducible  to  the  simple  (which  makes  them  demonstrable), 
(21.21).  The  whole  account  is  so  vague  that  we  are  not  in  a  position 
to  grasp  the  meaning  of  this  division.  But  in  Diogenes  we  find  a 
consistent  doctrine  of  these  same  'indemonstrable'  arguments; 
they  are  simply  the  axioms  of  the  Stoic  propositional  logic,  and  we 
consider  them  in  the  next  chapter. 


The  Stoics  made  clear  distinction  between  a  logical  rule  and  an 
instance  of  it,  i.e.  between  the  moods  (Tpo7roi)  of  an  argument 
and  the  argument  itself  -  a  distinction  which  Aristotle  applied  in 
practice,  but  without  a  theoretic  knowledge  of  it. 

21.22  These  are  some  of  the  arguments.  But  their  moods  or 
schemata  (ox^octoc)  in  which  they  are  formed  are  as  follows: 
of  the  first  indemonstrable:  'if  the  first,  then  (the)  second;  the 
first;  therefore  the  second';  of  the  second:  'if  the  first,  (then) 
the  second;  not  the  second;  therefore  not  the  first';  of  the 
third :  'not :  the  first  and  the  second ;  the  first;  therefore  not  the 

We  have  similar  schemata  for  other  arguments  as  well  (21.23  . 
even  for  some  of  the  not  indemonstrable  (cf.  22.17).  It  is  striking 
that  only  numerical  words  occur  in  them  as  variables.  One  might 
conjecture  that  this  was  so  in  Aristotle  too,  since  in  Greek  the 
letters  of  the  alphabet  could  function  as  numerals;  but  the  fact 



that  Aristotle  did  not  only  use  the  early  letters  of  the  alphabet, 
but  often  IT,  P,  and  2  as  well,  seems  to  exclude  this. 

Along  with  these  homogeneous  formulae  the  Stoics  also  had 
'mixed'  half-arguments,  half-schemata.  They  were  called  'argument- 
schemata'  (Xoy6Tpo7iot,). 

21.24  An  argument-schema  consists  of  both,  e.g.  'if  Plato 
lives,  Plato  breathes;  the  first;  therefore  the  second'.  The 
argument-schema  was  introduced  in  order  not  to  have  to 
have  a  long  sub-premiss  in  long  formulae,  so  as  to  gain  the 
conclusion,  but  as  short  as  possible:  'the  first;  therefore  the 

Another  example  is  this: 

21.25  If  the  sweat  flows  through  the  surface,  there  are 
intelligible  pores;  the  first;  therefore  the  second. 


The  Stoic  propositional  logic  seems  to  have  been  thoroughly 
axiomatized,  distinction  even  being  made  between  axioms  and 
rules  of  inference. 


The  tradition  is  obscure  about  the  axioms  (22.01 ;  et  vid.  supra 
21.21).  We  here  give  the  definition  of  the  indemonstrables  according 
to  Diogenes,  their  description,  with  examples,  from  Sextus. 

22.02  There  are  also  some  indemonstrables  (av<x7c68eixToi) 
which  need  no  demonstration,  by  means  of  which  every  (other) 
argument  is  woven;  they  are  five  in  number  according  to 
Chrysippus,  though  other  according  to  others.  They  are 
assumed  in  conclusives,  syllogisms  and  hypotheticals  (Tp07ux£>v) 

22.03  The  indemonstrables  are  those  of  which  the  Stoics 
say  that  they  need  no  proof  to  be  maintained.  .  .  .  They 
envisage  many  indemonstrables,  but  especially  five,  from 
which  it  seems  all  others  can  be  deduced. 

This  is  no  less  than  an  assertion  of  the  completeness  of  the  system : 
whether  it  is  correct  we  cannot  tell,  since  we  do  not  know  the 
metatheorems  and  have  only  a  few  of  the  derivative  arguments. 



22.04  The  first  (indemonstrable)  from  a  connected  (pro- 
position) and  its  antecedent  yields  its  consequent,  e.g.  'if 
it  is  day,  it  is  light;  it  is  day;  therefore  it  is  light'; 

22.05  the  second  from  a  connected  (proposition)  and  the 
contradictory  opposite  (avTixst^evou)  of  its  consequent  yields 
the  contradictory  opposite  of  its  antecedent,  e.g.  'if  it  is 
day,  it  is  light;  it  is  not  light;  therefore  it  is  not  day' ; 

22.06  the  third  from  the  negation  (obccxpaTixoO)  of  a  con- 
junction together  with  one  of  its  components,  yields  the 
contradictory  opposite  of  the  other,  e.g.  'not:  it  is  day  and 
it  is  night;  it  is  day;  therefore  it  is  not  night' ; 

22.07  the  fourth,  from  a  (complete)  disjunctive  (proposi- 
tion) together  with  one  of  the  (propositions)  disjoined  (e7te£evy- 
(jtivoov)  in  it,  yields  the  contradictory  opposite  of  the  other,  e.g. 
'either  it  is  day  or  it  is  night;  it  is  day;  therefore  it  is  not 
night' ; 

22.08  the  fifth  from  a  (complete)  disjunctive  (proposition) 
together  with  the  contradictory  opposite  of  one  of  the  dis- 
joined (propositions)  yields  the  other,  e.g.  'either  it  is  day  or  it 
is  night;  it  is  not  night;  therefore  it  is  day'. 

Other  less  reliable  sources  speak  of  two  further  indemonstrables, 
the  sixth  and  seventh  (22.09). 


The  reduction  of  demonstrable  arguments  to  indemonstrable,  was 
effected  in  Stoic  logic  by  means  of  certain  metalogical  rules.  One 
name  for  such  was  Os(jia,  but  it  seems  that  the  expression  OscopTjpia 
was  also  used  (22.10).  We  shall  call  them  'metatheorems',  in  accord- 
ance with  modern  usage.  A  text  of  Galen  shows  that  there  were 
at  least  four  of  them  (22.11),  but  only  the  first  and  third  are  stated 

22.12  There  is  also  another  proof  common  to  all  syllogisms, 
even  the  indemonstrable,  called  '(reduction)  to  the  impossible' 
and  by  the  Stoics  termed  'first  metatheorem'  (constitutio)  or 
'first  exposition'  (expos Hum).  It  is  formulated  thus:  'If  some 
third  is  deduced  from  two,  one  of  the  two  together  with  the 
opposite  of  the  conclusion  yields  the  opposite  of  the  other.' 

This  is  the  rule  for  reduction  to  the  impossible  (16.33),  already 
stated  by  Aristotle  in  another  form. 

22.13  The  essentials  of  the  so-called  third  metatheorem 
(SepuxTCK;)  look  like  this :  if  some  third  is  deduced  from  two  and 



one  (of  the  two)  can  be  deduced  syllogistically  from  others, 
the  third  is  yielded  by  the  rest  and  those  others. 

This  metatheorem  is  what  in  fact  underlies  the  aristotelian 
'direct  reduction'  of  syllogisms,  and  can  be  formulated: 

22.131  If  r  follows  from  p  and  q,  and  p  from  s,  then  r 
follows  from  q  and  s  (cf.  14.141). 

The  following  is  given  by  Alexander  as  the  'synthetic  theorem' 
(cjuvOstixov  0s:copY][jLa) : 

22.14  If  some  (third)  is  deduced  from  some  (premisses),  and 
if  the  deduced  (third)  together  with  one  or  more  (fourth) 
yields  some  (fifth),  then  this  (fifth)  is  deduced  also  from  those 
(premisses)  from  which  this  (third)  is  deduced. 

The  rule  being  stated  is  this: 

22.141  If  r  follows  from  p  and  g,  and  t  from  r  and  s, 
then  i  follows  from  p,  q  and  s; 

or,  if  one  represents  the  premisses  with  a  single  variable : 

22.142  If  q  follows  from  p,  and  s  from  q  and  r,  then  s 
follows  also  from  p  and  r. 

Sextus  cites  a  similar  but  seemingly  different  metatheorem: 

22.15  It  should  be  known  that  the  following  dialectical 
theorem  (OscopY^oc)  has  been  handed  down  for  the  analysis  of 
syllogisms :  'if  we  have  premisses  to  yield  a  conclusion,  then  we 
have  this  conclusion  too  potentially  among  these  (premisses), 
even  if  it  is  not  explicitly  (xoct*  excpopav)  stated. 

We  have  two  detailed  examples  of  the  apllication  of  this  meta- 
theorem, which  belong  to  the  highest  development  of  Stoic  logic. 


22.16  Of  the  not-simple  (arguments)  some  consist  of 
homogeneous,  others  of  not  homogeneous  (arguments).  Of 
not  homogeneous,  those  which  are  compounded  of  two  first 
indemonstrables  (22.04),  or  of  two  second  (22.05).  Of  not 
homogeneous,  those  which  (are  compounded)  of  second  and 
third*  (22.06),  or  in  general  of  such  (dissimilars).  An  example  of 

*   Reading  xal  Tptxou  in  the  lacuna  with  Kochalsky. 



those  consisting  of  homogeneous  (arguments)  is  the  following: 
'if  it  is  day*,  then  if  it  is  day  it  is  light;  it  is  day;  therefore  it 
is  light'.  .  .  .  For  we  have  here  two  premisses,  (1)  the  connected 
proposition:  'if  it  is  day*  *,  then  if  it  is  day  it  is  light',  which 
begins  with  the  simple  proposition  'it  is  day'  and  ends  with 
the  not  simple,  connected  proposition  'if  it  is  day  it  is  light'; 
and  (2)  the  antecedent  in  this  (first  premiss:)  'it  is  day'.  If 
by  means  of  the  first  indemonstrable  we  infer  from  those  the 
consequent  of  the  connected  (proposition,  viz.)  'if  it  is  day  it 
is  light',  then  we  have  this  inferred  (proposition)  potentially  in 
the  argument,  even  though  not  explicitly  stated.  Putting  this 
now  together  with  the  minor  premiss  of  the  main  argument, 
viz.  'it  is  day',  we  infer  by  means  of  the  first  indemonstrable: 
'it  is  light',  which  was  the  conclusion  of  the  main  argument. 

22.17  That  is  what  the  arguments  are  like  which  are  com- 
pounded from  homogeneous  (indemonstrables).  Among  the 
not  homogeneous  is  that  propounded  by  Ainesidemus  about 
the  sign,  which  runs  thus:  'if  all  phenomena  appear  similarly 
to  those  who  are  similarly  disposed,  and  signs  are  phenomena, 
then  signs  appear  similarly  to  all  those  who  are  similarly 
disposed;  signs  do  not  appear  similarly  to  all  those  who  are 
similarly  disposed;  phenomena  appear  similarly  to  all  those 
who  are  similarly  disposed ;  therefore  signs  are  not  phenomena'. 
This  argument  is  compounded  of  second  and  third  indemon- 
strables, as  we  can  find  out  by  analysis.  This  will  be  clearer  if 
we  put  the  process  in  the  form  of  the  schema  of  inference: 
'if  the  first  and  second,  then  the  third ;  not  the  third ;  the  first; 
therefore  not  the  second'.  For  we  have  here  a  connected  (propo- 
sition) in  which  the  conjunction  of  the  first  and  second  forms 
the  antecedent,  and  the  third  the  consequent,  together  with 
the  contradictory  opposite  of  the  consequent,  viz.  'not  the 
third'.  Hence  we  infer  by  means  of  the  second  indemonstrable 
the  contradictory  opposite  of  the  antecedent,  viz.  'therefore 
not:  the  first  and  the  second'.  But  this  is  potentially  contained 
in  the  argument,  as  we  have  it  in  the  premisses  which  yield  it, 
though  not  verbally  expressed.  Putting  it  *  *  *  together  with  the 
other  premiss,  the  first,  we  infer  the  conclusion  (of  the  main 
argument),  'not  the  second',  by  means  of  the  third  indemon- 

*   Adding  with  Kochalsky:  si  yjfiipa  saxtv. 
*  *   Completing  the  text  as  before. 
*  *  *   reading  07rsp  instead  of  a7rep,  with  Kochalsky. 



According  to  Cicero  (22.18)  the  Stoics  derived  'innumerable' 
arguments  in  similar  ways. 

22.19  The  said  (Chrysippus)  says  that  it  (the  dog)  often* 
applies  the  fifth  indemonstrable,  when  on  coming  to  the 
meeting  of  three  roads  it  sniffs  at  two  down  which  the  game 
has  not  gone  and  immediately  rushes  down  the  third  without 
sniffing  at  it.  The  sage  says  in  fact  that  it  virtually  infers: 
the  game  has  gone  down  this,  or  this,  or  that;  neither  this  nor 
this;  therefore  that. 

22.21  If  two  connected  (propositions)  end  in  contradictorily 
opposed  (consequents)  -  this  theorem  is  called  (the  theorem) 
'from  two  connecteds'  (Tpo7cixc5v)  -  the  (common)  antecedent 
of  the  two  is  refuted  ....  This  argument  is  formed  according  to 
the  schema  of  inference :  'if  the  first,  the  second ;  if  the  first  *  *, 
not  the  second;  therefore  not  the  first'.  The  Stoics  give  it 
material  expression  (i.e.  by  a  substitution)  when  they  say 
that  from  the  (proposition)  'if  you  know  that  you  are  dead 
(you  are  dead  if  you  know  that  you  are  dead)  you  are  not  dead' 
there  follows  this  other:  'therefore  you  do  not  know  that  you 
are  dead'. 

22.22  Some  argue  in  this  way:  'if  there  are  signs,  there  are 
signs;  if  there  are  not  signs,  there  are  signs;  there  are  either  no 
signs  or  there  are  signs;  therefore  there  are  signs'. 

§  23.  THE  LIAR 

The  Stoics  and  above  all  the  Megarians  devoted  much  attention 
to  fallacies.  Some  of  the  ones  they  considered  derive  from  the  problem 
of  the  continuum  and  belong  to  mathematics  in  the  narrower  sense 
of  that  word ;  the  rest  are  mostly  rather  trifles  than  serious  logical 
problems  (23.01).  But  one  of  their  fallacies,  'the  Liar'  (^£uSo[X£vo<;) 
has  very  considerable  logical  interest  and  has  been  deeply  studied 
by  logicians  for  centuries,  in  antiquity,  the  middle  ages,  and  the 
20th  century.  The  Liar  is  the  first  genuine  semantic  fallacy  known  to 

*  Sia  tcXeiovcov:  this  could  also  mean  'the  (argument)  from  the  more';  but  I 
follow  Mates  (22.20)  since  (1)  we  know  of  no  such  indemonstrable,  and  (2)  the 
argument  is  reducible  to  the  simple  fifth  indemonstrable. 
*  *   omitting  ou  with  Koetschau. 



In  St.  Paul  is  to  be  found  the  following  notable  text: 

23.02  One  of  themselves,  a  spokesman  of  their  own,  has 
told  us :  The  men  of  Crete  were  ever  liars,  venomous  creatures, 
all  hungry  belly. 

According  to  various  sources  (23.03)  this  spokesman  was  Epimeni- 
des,  a  Greek  sage  living  at  the  beginning  of  the  6th  century  b.c. 
Hence  the  Liar  is  often  called  after  him,  but  wrongly,  for  Epimenides 
was  clearly  not  worrying  about  a  logical  paradox.  Plato,  too,  who  con- 
sidered similar  problems  in  the  Euthydemus  (ca.  387  B.C.;  23.04;  did 
not  know  the  Liar.  But  Aristotle  has  it  in  the  Sophistic  Refutations, 
about  330  b.c.  (23.05).  Now  that  is  just  the  period  when  Eubulides 
was  flourishing,  to  whom  Diogenes  Laertius  explicitly  ascribes  the 
discovery  of  the  Liar  (18.02).  After  that,  Theophrastus  wrote  three 
books  on  the  subject  (23.06),  Chrysippus  many  more,  perhaps 
twenty-eight  (23.07).  How  much  people  took  the  problem  to  heart 
at  that  time  can  be  seen  from  the  fact  that  one  logician,  Philetas  of 
Cos  (ca.  340-285  b.c),  died  because  of  it: 

23.08  Traveller,  I  am  Philetas;  the  argument  called  the 
Liar  and  deep  cogitations  by  night,  brought  me  to  death. 


In  spite  of  this  interest  and  the  extensive  literature  about  the 
Liar,  we  no  longer  possess  Eubulides's  formulation  of  the  antinomy, 
and  the  versions  that  have  come  down  to  us  are  so  various  that  it 
is  impossible  to  determine  whether  a  single  formula  underlies  them 
all,  and  which  of  the  surviving  ones  has  been  considered  by  com- 
petent logicians.  Here  we  can  only  give  a  simple  list  of  the  most 
important,  collected  by  A.  Riistow  (23.09).  They  seem  to  fall  into 
four  groups. 


23.10  If  you  say  that  you  lie,  and  in  this  say  true,  do  you 
lie  or  speak  the  truth  ? 

23.11  If  I  lie  and  say  that  I  lie,  do  I  lie  or  speak  the  truth? 


23.12  If  you  say  that  you  lie,  and  say  true,  you  lie;  but 
you  say  that  you  lie,  and  you  speak  the  truth;  therefore  you 

23.13  If  you  lie  and  in  that  say  true,  you  lie. 



23.14  I  say  that  I  lie,  and  (in  so  saying)  lie;  therefore  I 
speak  the  truth. 

23.15  Lying,  I  utter  the  true  speech,  that  I  lie. 


23.16  If  it  is  true,  it  is  false ;  if  it  is  false,  it  is  true. 

23.17  Whoso  says  'I  lie',  lies  and  speaks  the  truth  at  the 
same  time. 

The  relation  of  the  four  groups  to  one  another  is  as  follows :  the 
texts  of  the  first  group  simply  posit  the  question:  is  the  Liar  true  or 
false  ?  Those  of  the  second  conclude  that  it  is  true,  of  the  third  that 
it  is  false.  The  texts  of  the  fourth  group  draw  both  conclusions 
together;  the  proposition  is  both  true  and  false. 


Aristotle  deals  with  the  Liar  summarily  in  that  part  of  his 
Sophistic  Refutations  in  which  he  discusses  fallacies  dependent  on 
what  is  said  'absolutely  and  in  a  particular  respect': 

23.18  The  argument  is  similar,  also,  as  regards  the  problem 
whether  the  same  man  can  at  the  same  time  say  what  is  both 
false  and  true:  but  it  appears  to  be  a  troublesome  question 
because  it  is  not  easy  to  see  in  which  of  the  two  connections 
the  word  'absolutely'  is  to  be  rendered  -  with  'true'  or  with 
'false'.  There  is,  however,  nothing  to  prevent  it  from  being 
false  absolutely,  though  true  in  some  particular  respect  (nfi)  or 
relation  (tiv6<;),  i.e.  being  true  in  some  things  though  not 
true  absolutely. 

It  has  been  said  (23.19)  that  the  difficulty  is  here  'quite  unresolved, 
and  indeed  unnoticed',  and  indeed  Aristotle  has  not  solved  our 
antinomy  nor  understood  its  import.  Yet,  as  is  so  often  the  case 
with  this  pastmaster,  he  reveals  a  penetrating  insight  into  the 
principle  of  the  medieval  and  modern  solutions  -  the  necessity  of 
distinguishing  different  aspects,  levels  as  we  now  say,  in  the  Liar. 
Worth  noting  too,  is  Aristotle's  standpoint  of  firm  conviction  that 
a  solution  is  discoverable.  This  conviction  has  remained  the  motive 
power  of  logic  in  this  difficult  field. 

The  solution  of  Chrysippus  has  reached  us  in  a  very  fragmentary 
papyrus,  written  moreover  in  difficult  language.  Its  essential,  and 
most  legible,  part  is  as  follows* : 

*  Thanks  are  due  to  Prof  O.  Gigon  for  help  with  this  text. 



23.20  The  (fallacy)  about  the  truth-speaker  and  similar 
ones  are  to  be  .  .  .  (solved  in  a  similar  way).  One  should  not  say 
that  they  say  true  and  (also)  false;  nor  should  one  conjecture 
in  another  way,  that  the  same  (statement)  is  expressive  of  true 
and  false  simultaneously,  but  that  they  have  no  meaning  at  all. 
And  he  rejects  the  afore-mentioned  proposition  and  also  the 
proposition  that  one  can  say  true  and  false  simultaneously  and 
that  in  all  such  (matters)  the  sentence  is  sometimes  simple, 
sometimes  expressive  of  more. 

The  most  important  words  in  this  text  are  cn)[>  ziXzac, 
(X7ro7rAavc5vTai,  translated  '(that)  they  have  no  meaning  at  all'. 
The  Greek  phrase  is  ambiguous  as  between  (1)  that  whoever  states 
the  Liar  attributes  a  false  assertion  to  the  proposition,  and  (2)  that 
he  says  something  which  has  no  meaning  at  all.  The  fragmentary 
context  seems  to  indicate  the  second  interpretation  as  the  correct 
one,  but  it  is  impossible  to  be  certain  of  this.  If  it  is  correct,  Ghrysip- 
pus's  solution  is  that  the  Liar  is  no  proposition  but  a  senseless 
utterance,  which  would  be  a  view  of  the  highest  importance.  The 
Aristotelian  attempt  to  solve  it  is  definitely  rejected  in  this  text. 





With  the  end  of  the  old  Stoa  there  begins  a  period  into  which 
hardly  any  research  has  been  done.  However,  on  the  basis  of  the 
few  details  known  to  us  we  may  suppose  with  great  probability  that 
the  formal  logic  of  this  period  was  of  the  following  kind : 

1.  The  period  is  not  a  creative  one.  No  new  problems  or  original 
methods  such  as  those  developed  by  Aristotle  and  the  Megarian- 
Stoic  school  are  to  be  found. 

2.  Yet,  apparently  right  up  to  the  fall  of  the  Roman  empire, 
individual  scholarly  works  appeared.  Some  earlier  methods  were 
improved,  the  material  was  systematized  and  sometimes  developed. 
There  were  even  not  wanting  genuinely  gifted  logicians,  among  the 
best  of  whom  was  Alexander  of  Aphrodisias. 

3.  The  logical  literature  consisted  chiefly  of  two  kinds  of  work: 
big  commentaries,  mainly  on  Aristotle,  and  handbooks. 

4.  As  to  their  content,  we  discern  mostly  a  syncretizing  tendency 
in  the  sense  that  Aristotelian  and  Stoic-Megarian  elements  are 
mingled,  Stoic  methods  and  formulations  being  applied  to  Aristo- 
telian ideas. 

Lack  of  monographs  makes  it  impossible  to  survey  the  state  of  the 
logical  problematic  during  the  period,  and  we  limit  ourselves  to  the 
choice  of  some  particular  doctrines  so  far  found  in  the  mass  of 
commentaries  and  handbooks.  But  first  some  of  the  most  important 
thinkers  must  be  named. 

The  first  well-known  logicians  of  this  period  are  Galen  and  the 
less  notable  Apuleius  of  Madaura  whose  handbooks  have  survived; 
the  former  is  the  subject  of  the  only  monograph  on  the  period 
(24.01).  In  the  3rd  century  a.d.  we  find  Alexander  of  Aphrodisias, 
already  mentioned,  one  of  the  best  commentators  on  the  whole 
Aristotelian  logic,  and  unlike  Galen  and  Apuleius  a  fairly  pure 
Aristotelian.  Porphyry  of  Tyre  lived  about  the  same  time,  and 
composed  an  Introduction  (ziGOLycxtyri)  to  the  Aristotelian  categories. 
In  it  he  systematized  the  doctrine  of  the  predicables  (11.06ff.), 
giving  a  five-fold  enumeration:  genus,  specific  difference,  species, 
property  and  accident  (24.02).  This  work  was  to  be  basic  in  the 
Middle  Ages.  Later  logicians  include  Iamblichus  of  Chalcis,  not  to 
be  taken  very  seriously,  Themistius  (both  these  in  the  4th  century 
a.d.),  Ammonius  Hermeae  (5th  century),  Martianus  Capella,  author 



of  a  handbook  which  formed  an  important  link  between  ancient 
and  later  logic  (5th  century),  AmmoniuB  the  Peripatetic,  Simplicius 
(6th  century),  who  was  another  of  the  better  commentators  on 
Aristotle,  and  finally  Philoponus  (7th  century),  but  these  have 
little  importance  so  far  as  we  can  judge.  On  the  other  hand  the  last 
Roman  logician,  Boethius  (ca.  480-524)  is  of  fairly  considerable 
importance  both  because  his  works  became  a  prime  source  for  the 
Scholastics  and  also  because  he  transmits  doctrines  and  methods 
not  mentioned  elsewhere,  though  he  himself  was  only  a  moderate 
logician.  With  his  execution  the  West  enters  on  a  long  period  without 
any  logic  worth  speaking  of. 


Of  the  commentators'  discoveries  the  'tree  of  Porphyry'  has 
certainly  achieved  the  greatest  fame.  While  it  can  be  regarded  as 
only  a  compendium  of  Aristotelian  doctrines  it  has  great  importance 
as  comprising  (1)  a  system  of  classification,  which  was  not  to  the 
fore  in  Aristotle's  thought  (11.13),  and  (2)  an  extensional  view  of 
terms.  First  we  give  the  text: 

24.03  Let  what  is  said  in  one  category  now  be  explained. 
Substance  (ouctioc)  is  itself  a  genus,  under  this  is  body,  and  under 
body  is  living  (s^uxov)  body,  under  which  is  animal.  Under 
animal  is  rational  (Xoyixov)  animal,  under  which  is  man.  Under 
man  are  Socrates  and  Plato  and  individual  (xorra  uipo<;)  men. 
But  of  these,  substance  is  the  most  generic  and  that  which  is 
genus  alone ;  man  is  the  most  specific  and  that  which  is  species 
alone.  Body  is  a  species  of  substance,  a  genus  of  living  body. 

The  following  text  shows  how  thoroughly  extensional  a  view  is 
being  taken: 

24.04  (Genus  and  species)  differ  in  that  genus  contains 
(■7T£ptix£L)  its  species,  the  species  are  contained  in  but  do  not 
contain  their  genus.  For  the  genus  is  predicated  of  more  things 
than  the  species. 

This  conception  is  carried  so  far  that  one  can  here  properly  speak 
of  a  beginning  of  calculus  of  classes.  At  the  same  time  Porphyry 
makes  a  distinction  which  corresponds  fairly  closely  to  the  modern 
distinction  between  extension  and  intension  (36.10,  45.03)  -  or. 
again,  between  simple  and  personal  supposition  (27.15).  For  among 
a  number  of  definitions  of  the  predicables,  he  has : 



24.05  The  philosophers  .  .  .  define,  saying  that  genus  is  what 
is  predicated  essentially  (ev  tw  tl  eoti)  of  several  things 
differing  in  species. 

24.06  The  genus  differs  from  the  difference  and  the  common 
accidents  in  that,  while  the  difference  and  the  common  acci- 
dents are  predicated  of  several  things  differing  in  species,  they 
are  not  predicated  essentially  but  as  qualifying  (ev  tco  o7rot6v  tl 
e<mv).  For  when  we  ask  what  it  is  of  which  these  are  predicated, 
we  answer  with  the  genus;  but  we  do  not  answer  with  the 
differences  or  accidents.  For  they  are  not  predicated  essentially 
of  the  subject  but  rather  as  qualifying  it.  For  on  being  asked 
of  what  quality  man  is,  we  say  that  he  is  rational,  and  to  the 
question  of  what  quality  crow  is,  we  answer  that  it  is  black. 
But  rational  is  a  difference,  and  black  an  accident.  But  when 
we  are  asked  what  man  is,  we  answer  that  he  is  animal,  animal 
being  a  genus  of  man. 


Among  the  most  important  achievements  of  this  period  are  two 
devices  which  so  far  as  we  know  were  unknown  to  Aristotle  and  the 
Stoics,  viz.  (1)  identification  of  variables,  (2)  substitution  of  sen- 
tential forms  for  variables. 

1.  Alexander  of  Aphrodisias 

The  first  is  to  be  found  in  Alexander  in  a  new  proof  of  the  con- 
vertibility of  universal  negative  sentences : 

24.07  If  someone  were  to  say  that  the  universal  negative 
(premiss)  does  not  convert,  (suppose)  A  belongs  to  no  B;  if 
(this  premiss)  does  not  convert,  B  belongs  to  some  A;  there 
results  in  the  first  figure  (the  conclusion  that)  to  some  A  A 
does  not  belong,  which  is  absurd. 

Alexander  here  makes  use  of  the  fourth  syllogism  of  the  first 
figure  (Ferio:  (13.06),  which  in  Aristotle's  presentation  runs:  'if  A 
belongs  to  no  B,  but  B  to  some  C,  then  to  some  C  A  cannot  belong'. 
He  identifies  C  with  A  -  i.e.  substitutes  one  variable  for  the  other, 
and  obtains:  'if  A  belongs  to  no  B,  but  B  to  some  A,  then  to  some  A 
A  does  not  belong'.  That  is  the  novelty  of  the  process. 

This  is  consonant  with  Alexander's  clear  insight  into  the  nature 
of  laws  of  formal  logic.  He  seems  to  have  been  the  first  to  make 
explicit  the  distinction  between  form  and  matter,  and  at  the  same 
time  to  have  come  close  to  an  explicit  determination  of  the  notion 
of  a  variable. 



24.08  He  (Aristotle)  introduces  the  use  of  letters  in  order 
to  show  us  that  the  conclusions  are  not  produced  in  virtue 
of  the  matter  but  in  virtue  of  such  and  such  a  form 
(oXWol)  and  composition  and  the  mood  of  the  premisses;  the 
syllogism  concludes  .  .  .  not  because  of  the  matter,  but  because 
the  formula  (au^uyta)  is  as  it  is.  The  letters  show  that  the 
conclusion  is  of  such  a  kind  universally  and  always  and  for 
every  choice  (of  material). 

2.   Boethius 

A  further  development  of  the  technique  of  formal  logic  is  to  be 
found  in  Boethius.  He  is  evidently  aiming  at  the  formulation  of  a 
rule  of  substitution  for  propositional  variables ;  this  is  not  given  in  the 
form  of  such  a  rule,  but  in  a  description  of  the  structure  of  formulae. 
Again  we  have  a  fairly  clear  distinction  between  form  and  matter  in 
a  proposition,  a  distinction  which  was  to  play  a  great  part  in  later 
history : 

24.09  We  shall  now  show  the  likenesses  and  differences 
between  simple  propositions  and  compound  hypothetical  ones. 
For  when  the  (hypothetical  propositions)  which  consist  of 
simple  ones  are  compared  with  those  which  are  compounded  of 
two  hypotheticals,  (one  sees  that)  the  sequence  (in  both  cases) 
is  the  same  and  the  relation  (of  the  parts  to  one  another) 
remains,  only  the  terms  are  doubled.  Since  the  places  which  are 
occupied  by  simple  propositions  in  those  hypotheticals  consist- 
ing of  simple  propositions,  are  occupied  in  hypotheticals  consist- 
ing of  hypotheticals  by  those  conditions  in  virtue  of  which  those 
(component)  propositions  are  said  to  be  joined  and  linked 
together.  For  in  the  proposition  which  says:  'if  A  is,  B  is', 
and  in  that  which  says:  'if,  if  A  is,  B  is,  (then)  if  C  is,  D  is' 
the  place  occupied  in  that  consisting  of  two  simple  propositions 
by  that  which  is  first:  'if  A  is',  in  the  proposition  consisting  of 
two  hypotheticals  is  occupied  by  that  which  (there)  is  first:  'if, 
if  A  is,  B  is'. 

If  we  remember  the  Stoic  distinction  between  argument  and  mood 
(21.22)  the  last  two  texts  do  not  seem  very  original;  but  they  are  the 
first  in  which  an  explicit  statement  of  the  distinction  is  found. 

It  is  Boethius  again  who  gives  a  fresh  division  of  implication : 

24.10  Every  hypothetical  proposition  is  formed  either  by 
connection  (connexionem)  ...  or  by  disjunction.  .  .  .  But  since 



it  has  been  said  that  the  same  thing  is  signified  by  the  connec- 
tives (conjunctione)  'si'  and  'cum'  when  they  are  put  in  hypo- 
thetical sentences,  conditionals  can  be  formed  in  two  ways: 
accidentally,  or  so  as  to  have  some  natural  consequence. 
Accidentally  in  this  way,  as  when  we  say:  'when  fire  is  hot, 
the  sky  is  round'.  For  the  sky  is  round  not  because  fire  is  hot, 
but  the  sentence  means  that  at  what  time  fire  is  hot,  the 
sky  is  round.  But  there  are  others  which  have  within  them 
a  natural  consequence,  .  .  .  e.g.  we  might  say:  'when  man  is, 
animal  is'. 

There  is  here,  as  often  elsewhere,  a  certain  obscurity  in  Boethius's 
thought  (24.11).  Apart  from  that,  his  division  of  implication  is 
something  of  a  backward  step  in  comparison  with  the  Stoic  dis- 
cussions of  the  subject  (vide  supra  20.05 ff.).  Yet  the  text  just  cited  is 
important  for  our  history,  being  an  evident  starting-point  for  scholas- 
tic speculations  about  implication. 

Hence  also  we  mention  the  following  details  of  Boethius's  doc- 
trine about  propositional  functors.  He  often  seems  to  use  lsV  (24.12) 
as  a  symbol  of  equivalence  (cf.  20.20 ff.).  The  sense  of  the  expression 
kauV  is  ambiguous.  On  the  one  hand,  we  find  -  and  for  the  first 
time  -  a  definition  in  the  sense  of  non-exclusive  alternation  (logical 
sum :  cf.  20.17,  30.18,  40.11,  41.18) : 

24.13  The  disjunctive  proposition  which  says  (proponit) : 
'either  A  is  not  or  B  is  not'  is  true  (fit)  of  those  things  which 
can  in  no  way  co-exist,  since  it  is  also  not  necessary  that 
either  one  of  them  should  exist;  it  is  equivalent  to  that  com- 
pound proposition  in  which  it  is  said :  'if  A  is,  B  is  not'.  ...  In 
this  proposition  only  two  combinations  yield  (valid)  syllogisms. 
For,  if  A  is,  B  will  not  be,  and  if  B  is,  A  will  not  be.  .  .  .  For 
if  it  is  said:  'either  A  is  not  or  B  is  not',  it  is  said:  'if  A  is,  B 
will  not  be',  and  'if  B  is,  A  will  not  be'. 

First  we  have  here  Sheffer's  functor  ('not  p  or  not  q'-,  43.43); 
secondly  this  text  contains  an  exact  definition  of  the  logical  sum. 
The  essential  idea  can  be  formulated : 

24.131  Not  p  or  not  q  if  and  only  if :  if  p  then  not  q. 

Putting  therein  'not-p'  for  'p'  and  'not-g'  for  lq\  we  get  by  the 
principle  of  double  negation: 

24.132  p  or  q  if  and  only  if:  if  not-p  then  q. 


On  the  other  hand,  Boethius  defines  in  analogous  fashion  his 

lsV  in  the  sense  of  equivalence  by  means  of  the  same  'auV  -  which 
therefore  and  in  this  case  has  the  sense  of  aegaied  equivalence 
(p  or  q  but  not  both,  and  necessarily  one  of  the  two)  (24.14). 

It  is  also  worth  remarking  that  Boethius  regularly  uses  the 
principle  of  double  negation  and  a  law  analogous  to  24.21. 


We  here  give  the  list  of  Boethius's  hypothetical  syllogisms.  They 
seem  to  be  the  final  result  of  Stoic  logic,  if  understood  as  laws  of  the 
logic  of  propositions.  Our  supposition  that  Boethius  aspired  to  a  rule 
of  substitution  for  propositional  variables  (cf.  24.09),  requires  them  to 
be  so  understood.  They  would  be  the  final  result  of  Stoic  logic  in  the 
sense  that  they  are  practically  the  only  part  of  this  logic  that  was 
preserved  by  Boethius  for  the  Middle  Ages. 

24.15  If  A  is,  B  is;  but  A  is;  therefore  B  is. 

24.16  If  A  is,  B  is;  but  B  is  not;  therefore  A  is  not. 

24.17  If  A  is,  B  is,  and  if  B  is,  C  must  be;  but  then:  if  A  is, 
C  must  be. 

24.18  If  A  is,  B  is,  and  if  B  is,  C  too  must  be;  but  C  is  not; 
therefore  A  is  not. 

24.19  If  A  is,  B  is ;  but  if  A  is  not,  C  is ;  I  say  therefore  that 
if  B  is  not,  C  is. 

24.20  If  A  is,  B  is  not;  if  A  is  not,  C  is  not;  I  say  therefore 
that  if  B  is,  C  is  not. 

24.21  If  B  is,  A  is;  if  C  is  not,  A  is  not;  on  this  supposition 
I  say  that  if  B  is,  it  is  necessary  that  C  is  not. 

24.22  If  B  is,  A  is;  if  C  is  not,  A  is  not;  I  say  therefore:  if 
B  is  is,  C  will  be. 

24.23  If  one  says :  'either  A  is  or  B  is',  (then)  if  A  is,  B  \x\\\ 
not  be;  and  if  A  is  not,  B  will  be;  and  if  B  is  not,  A  will  be; 
and  if  B*  is,  A  will  not  be. 

24.25  The  (proposition)  that  says:  'either  A  is  not  or  B  is 
not',  certainly  means  this,  that  if  A  is,  B  cannot  be. 

Boethius  developes  these  syllogisms  by  substituting  a  conditional 
proposition  for  one  or  both  variables  (cf.  24.09) ;  in  so  doing  he  treats 
the  negation  of  a  conditional  as  the  conjunction  of  the  antecedent 
with  the  negation  of  the  consequent,  according  to  the  law.  which  is 
not  expressly  formulated: 

24.251  Not:  if  p,  then  q,  if  and  only  if:  p  and  not-g. 

*  omitting  non  with  van  den  Driessche  (24.24). 



Finally  he  applies  the  law  of  double  negation  (cf.  20.041),  thus 
gaining  eighteen  more  syllogisms  (24.26). 



24.27  But  Ariston  the  Alexandrian  and  some  of  the  later 
Peripatetics  further  introduce  five  more  moods  (formed  from 
those)  with  a  universal  conclusion:  three  in  the  first  figure, 
two  in  the  second  figure,  which  yield  particular  conclusions. 
(But)  it  is  extremely  foolish  to  conclude  to  less  from  that  to 
which  more  is  due. 

This  text  is  not  very  clear.  But  its  difficulty  is  somewhat  lessened 
if  we  suppose  that  a  combination  of  two  Aristotelian  rules  is  envi- 
saged: (1)  that  allowing  a  universal  conclusion  to  be  weakened  to 
the  corresponding  particular  (13.23),  (2)  that  yielding  a  further 
conclusion  by  conversion  of  the  one  first  obtained.  Then  the  follow- 
ing would  be  the  moods  intended : 

24.271  A  to  all  B;  B  to  all  C;  therefore  A  to  some  C 

24.272  A  to  no  B;  B  to  all  C;  therefore  to  some  C,  A  not 

24.273  A  to  all  B;  B  to  some  C ;  therefore  C  to  some  A 

ZZk.Zlb  B  to  no  A;  B  to  all  C;  therefore  to  some  C,  A  not 

24.275  B  to  all  A;  B  to  no  C ;  therefore  to  some  C,  A  not 

Beyond  these,  Galen  transmits  a  further  mood  of  this  kind  in  the 
third  figure  (24.28): 

24.281  A  to  all  B;  C  to  all  B;  therefore  C  to  some  A 

These  formulae  all  have  a  Stoic  rather  than  an  Aristotelian  form. 
In  fact  from  Apuleius  on,  such  alteration  of  the  old  laws  into  rules 
is  more  or  less  standard  practice,  especially  in  Boethius. 

A  further  precision  given  to  the  Aristotelian  syllogistic  is  in  the 
famous  logical  square.  This  figure  is  first  found  in  Apuleius  again.  It 
looks  like  this: 










universal  affirmative 


universal  negal  ive 


all  pleasure  is  good 

no  pleasure  is  good 

some  pleasure  is  good 

some  pleasure  is  not  good 

particular  affirmative 

particular  negative 



sub  pares 


In  an  anonymous  fragment,  belonging  possibly  to  the  6th  cen- 
tury, we  read : 

24.30  Theophrastus  and  Eudemus  also  added  other  formu- 
lae to  those  of  Aristotle  in  the  first  figure  .  .  .  many  moderns 
have  thought  to  form  the  fourth  figure  therefrom,  citing 
Galen  as  the  author  of  this  intention. 

But  this  allegedly  'Galenic'  figure  is  not  to  be  found  in  him.  On 
the  contrary  he  plainly  states  that  there  are  only  three  figures : 

24.31  These  syllogisms  are  called,  as  I  have  said,  categorical ; 
they  cannot  be  formed  in  more  than  the  three  figures  men- 
tioned, nor  in  another  number  in  each  (of  these  figures);  this 
has  been  shown  in  the  treatises  On  Demonstration. 

J.  Lukasiewicz  was  able  to  explain  by  means  of  another  anony- 
mous fragment  how  nevertheless  the  discovery  of  the  fourth  figure 
could  be  credited  to  Galen  (24.32).  This  fragment  is  not  without 
historical  interest  even  apart  from  this  question: 

24.33  Of  the  categorical  (syllogism)  there  are  two  kinds; 
the  simple  and  the  compound.  Of  the  simple  syllogism  there 
are  three  kinds:  the  first,  the  second,  and  the  third  figure.  Of 
the  compound  syllogism  there  are  four  kinds:  the  first,  the 
second,  the  third,  and  the  fourth  figure.  For  Aristotle  says 
that  there  are  only  three  figures,  because  he  looks  at  the  simple 
syllogisms,  consisting  of  three  terms.  Galen,  however,  says 
in  his  Apodeidic  that  there  are  four  figures,  because  he  looks  at 
the  compound  syllogisms  consisting  of  four  terms,  as  he  has 
found  many  such  syllogisms  in  Plato's  dialogues. 



24.34  The  categorical  syllogism 

simple,  as  (  in)  Aristotle  compound,  as  (in)  Galen 

Figure  1,2,3 

Compound  figure 
1  to  1,  1  to  2,  1  to  3,  2  to  2,  2  to  1,  2  to  3,  3  to  3,  3  to  1,  3  to  2. 

1  to  1 


Compound  figure 

syllogistic : 

1  to  2  1  to  3 

2  3 

2  to  3 

2  to  2     3  to  3 
since  no  syllogism 
arises  from  two 
negatives  or  two 

1  to  1,  as  in  the  Alcibiades 

unsyllogistic : 

2  to  1     3  to  1     3  to  2 
2  3  4 


The  numerals  denote  the  successive  figures,  and  the  author 
means  that  a  valid  compound  syllogism  can  be  formed  in  four 
different  ways,  viz.  when  of  the  two  simple  syllogisms  from  which  it 
is  composed 

(1)  both  are  in  the  first  figure, 

(2)  the  first  is  in  the  first,  the  second  in  the  second  figure. 

(3)  the  first  is  in  the  first,  the  second  in  the  third, 

(4)  the  first  is  in  the  second,  the  second  in  the  third. 

Those  are  the  four  figures.  So  there  is  no  question  of  a  fourth  figure  of 
simple  syllogism,  which  was  only  ascribed  to  Galen  by  a  misunder- 
standing. Yet  the  unknown  scholiast  (24.30),  in  falling  a  victim  to 
this  misunderstanding  at  least  made  the  principle  of  the  fourth 
figure  another  interpretation  of  the  indirect  moods  of  Theophrastus 




Here  we  should  introduce  a  scheme  which  was  to  become  famous 
in  the  Middle  Ages  as  the  pons  asinorum  or  'asses'  bridge'  'I'l.Xil).  . 
It  is  to  be  found  in  Philoponus*,  and  is  an  elaboration  of  the 
Aristotelian  doctrine  of  the  inuentio  medii  (14.29;.  Although  it 
belongs  to  methodology  rather  than  logic,  it  is  relevant  to  the  latter 
also.  The  scheme  seems  typical  of  the  way  in  which  the  commentators 
developed  the  syllogistic.  In  Philiponus  the  lines  are  captioned  in  the 
figure  itself.  For  graphical  reasons  we  put  these  comments  after- 
wards and  refer  to  them  by  numbers. 




What  follows  on  the  good 

helpful,  eligible, 

to  be  pursued, 

suitable,  desirable, 



What  is  alien  to  the  good:  -4 
imperfect,  to  be 
fled  from,  harmful, 
bad,  ruinous,  alien, 

What  the  good 
follows  upon: 
happiness,  natural 
well-being,  final 
cause,  perfect, 
virtuous  life. 

What  follows  on 

movement,  natural 
activity,  unimpeded 
life,  object 
of  natural  desire, 

What  is  alien  to 
disease,  labour, 
fear,  need, 
unnatural  movement. 

What  pleasure 
follows  upon: 
health,  good 
repute,  virtuous 
life,  plenty, 
good  children,  freedom 
from  pain,  comfort. 

1)  Unsyllogistic,  because  of  concluding  in  the  second  figure 
from  two  universal  affirmative  (premisses). 

2)  Universal  negative   (conclusion)   in  the  first  and  second 
(figures)  by  two  conversions. 

3)  Particular  affirmative  (conclusion  in  the  first  and  third) 
figures  by  conversion  of  the  conclusion. 

4)  Universal  negative   (conclusion)   in  the  first  and  second 

*  Thanks  are  due  to  Prof.  L.  Minio-Paluello  for  pointing  out  this  passage. 
*  *  For  typographical  reason  the  words  in    the  figure  are  set  in  small  type, 
though  they  belong  to  the  quotation. 



5)  Universal  affirmative  (conclusion)  in  the  first  figure. 

6)  Unsyllogistic,  from  two  universal  negatives. 

7)  Particular  negative  (conclusion)  in  the  third  and  first 
(figure)  through  conversion  of  the  minor  (premiss). 

8)  Unsyllogistic,  since  the  particular  does  not  convert,  and  in 
the  first  (figure)  because  (the  syllogism)  has  a  negative  minor 

9)  Particular  affirmative  (conclusion)  in  the  third  and  first 
figures  by  conversion  of  the  minor  premiss. 


Finally  we  shall  speak  of  a  detail  which  had  no  influence  on  the 
later  development  of  logic,  but  which  yet  may  be  reckoned  an 
ingenious  anticipation  of  the  logic  of  relations.  Galen,  dividing 
syllogisms  in  his  Introduction,  distinguishes  first  between  categorical 
and  hypothetical  syllogisms,  thus  separating  term-  and  class-logic; 
he  then  adds  a  further  class: 

24.36  There  is  still  a  further,  third  class  of  syllogisms, 
useful  for  demonstration,  which  I  characterize  as  based  on 
relation.  Aristotelians  claim  that  they  are  counted  as  cate- 
goricals.  They  are  not  a  little  in  use  among  the  Sceptics, 
Arithmeticians  and  experts  in  calculation  in  certain  arguments 
of  this  kind:  'Theon  possesses  twice  what  Dion  possesses;  but 
Philo  too  possesses  twice  what  Theon  possesses;  therefore 
Philo  possesses  four  times  what  Dion  possesses.' 

This  is  in  fact  a  substitution  in  a  law  of  the  logic  of  relations,  and 
it  is  remarkable  that  Galen  divides  his  logic  just  as  Whitehead  and 
Russell  were  to  do  in  the  20th  century.  The  content  of  his  logic  of 
relations  is  of  course  very  poor,  and  he  thinks  that  such  laws  are 
reducible  to  categorical  syllogisms  (24.37),  which  is  a  regress  from 
the  position  of  Aristotle. 


To  summarize  the  results  of  post-Aristotelian  antiquity  we  can 

1.  Propositional  logic  was  then  created.  Some  theorems  of  this 
kind  were  already  known  to  Aristotle,  sometimes  even  stated  with 
propositional  variables:  but  these  were  rather  obiter  dicta  than 
systematically  presented.  In  the  Stoics  on  the  other  hand  we  meet 
systematic  theory  developed  for  its  own  sake. 

2.  This  system  is  based  on  a  fairly  well  worked  out  semantics, 
and  it  was  expressly  stated  in  the  Stoic  school  that  it  was  concerned 



neither  with  words  nor  psychic  images,  hut  with  objective  meanings, 
the  lecla.  We  have  therefore  to  thank  them  for  a  fundamental 
thesis  which  was  to  play  a  great  part  in  the  history  of  logic. 

3.  Megarian-Stoic  logic  contained  an  astonishingly  exact  analysis 
of  proposition-forming  functors:  we  find  correctly  formed  truth-tables 
and  a  more  intricate  discussion  of  the  meaning  of  implication  than 
we  seem  yet  to  have  attained  in  the  20th  century. 

4.  In  this  period  the  method  is  formalistic.  Unambiguous 
correlation  of  verbal  forms  to  lecla  being  presupposed,  attention  is 
exclusively  directed  to  the  syntactical  structure  of  expressions. 
The  application  of  this  method  and  the  logical  subtlety  shown  by  the 
Stoics  must  be  deemed  quite  exemplary. 

5.  This  formalism  is  accompanied  by  a  significant  extension  of 
logical  technique,  shown  in  the  clear  distinction  between  propositional 
functions  and  propositions  themselves,  the  method  of  identification 
of  variables,  and  the  application  of  the  rule  permitting  substi- 
tution of  propositional  functions  for  propositional  variables. 

6.  Propositional  logic  is  axiomatized,  and  a  clear  distinction 
drawn  between  laws  and  melalheorems. 

7.  Finally  we  have  to  thank  the  Megarian  school  for  propound- 
ing the  first  important  logical  antinomy  -  the  Liar  -  which  for 
centuries  remained  one  of  the  chief  problems  of  formal  logic,  and  is 
so  even  today.  So  without  exaggeration  one  can  say  that  the  achieve- 
ments of  this  period  make  up  antiquity's  second  basic  contribution 
to  formal  logic. 



The  Scholastic  Variety  of  Logic 



At  the  present  time  much  less  is  known  about  the  history  of 
scholastic  than  of  ancient  logic.  The  reason  is  that  when  Scholasti- 
cism ceased  to  be  disparaged  at  the  end  of  the  nineteenth  century, 
there  was  at  first  little  revival  of  interest  in  its  formal  logic.  This 
lack  of  interest  is  shown  in  the  fact  that  of  more  than  ten  thousand 
titles  of  recent  literature  on  Thomas  Aquinas  (up  to  1953),  very 
few  concern  his  formal  logic.  There  are  indeed  earlier  works  treating 
of  questions  of  the  literary  history  of  scholastic  logic  -  Grabmann 
having  done  most  to  find  and  publish  texts  -,  but  the  investigation 
of  their  logical  content  only  began  with  Lukasiewicz's  paper  of 
1934  (25.01),  pioneering  in  this  field  too.  Under  his  influence  some 
notable  medievalists,  e.g.  besides  Grabmann,  K.  Michalski,  applied 
themselves  to  logical  problems,  and  from  his  school  there  came  the 
first  work,  well  and  systematically  prepared,  on  medieval  logic,  the 
paper  on  the  propositional  logic  of  Ockham  by  J.  Salamucha  (1935) 
(25.02).  A  number  of  texts  and  treatises  followed,  those  of  Ph.  Boeh- 
ner  O.F.M.  and  E.  Moody  in  the  forefront.  Today  there  is  quite  a 
group  at  work,  though  as  yet  a  small  one. 

But  we  are  still  at  the  beginning.  Arabian  and  Jewish  logic  has 
hardly  been  touched ;  texts  and  treatises  are  alike  lacking.  In  the 
western  domain  some  texts  of  Abelard  have  been  published  for  the 
12th  century;  for  the  14th  and  15th  centuries  we  have  hardly 
anything,  either  new  editions  of  texts  or  works  on  them;  the  13th 
century  is  almost  completely  inaccessible  and  unknown.  For  this 
last,  besides  the  (fairly)  reliable  older  editions  of  the  works  of 
Thomas  Aquinas  and  (some  works)  of  Duns  Scotus,  Peter  of  Spain 
and  William  of  Shyreswood  are  available  only  in  provisional  editions. 
For  the  14th  century  there  is  an  edition  of  the  first  book  of  Ockham's 
Summa,  and  one  of  a  small  work  ascribed  to  Burleigh.* 

Altogether  we  must  say  that  the  present  state  of  research  permits 
no  general  survey  of  the  sources,  growth  and  details  of  scholastic 


However,  on  the  basis  of  the  works  of  Ph.  Boehner,  E.  Moody, 
L.  Minio-Paluello,  and  of  the  ever-growing  number  of  general 
studies  of  medieval  philosophy,  the  history  of  medieval  logic  can  be 
provisionally  divided  into  the  following  periods : 

*   The  late  Fr.  Ph.  Boehner  was  working  on  a  critical  edition  of  another  work 
of  Burleigh's  and  of  the  Perutilis  Logica  of  Albert  of  Saxony. 



1.  transitional  period:  up  to  Abelard.  So  far  as  we  know  this  is 
not  remarkable  for  any  logical  novelties,  and  acquaintance  even  with 
earlier  achievements  was  very  limited. 

2.  creative  period:  beginning  seemingly  after  Abelard,  about 
1150,  and  lasting  to  the  end  of  the  13th  century.  Former  achievement- 
now  became  known  in  the  West,  partly  through  the  Arabs,  partly  (as 
L.  Minio-Paluello  has  shown*)  directly  from  Byzantium.  At  the 
same  time  work  began  on  new  problems,  such  as  the  proprieties 
lerminorum,  properties  of  terms.  By  about  1260  the  essentials  of 
scholastic  logic  seem  to  have  taken  shape  and  been  made  widely 
known  in  text-books.  The  best  known  book  of  this  kind,  and  the 
most  authoritative  for  the  whole  of  Scholasticism  -  though  by  no 
means  the  first  or  the  only  one  -  is  the  Summulae  Logicales  of 
Peter  of  Spain. 

3.  period  of  elaboration:  beginning  approximately  with  William 
of  Ockham  (ob.  1349/50)**  and  lasting  till  the  close  of  the  Middle 
Ages.  No  essentially  new  problems  were  posed,  but  the  old  were 
discussed  very  thoroughly  and  very  subtly,  which  resulted  in  an 
extremely  comprehensive  logic  and  semiotic. 

So  little  is  known  of  the  whole  development  that  we  are  unable 

even  to  name  only  the  most  important  logicians.  We  can  only  say 

with   certainty  that  the   following  among  others   exercised   great 

influence : 

in  the  12th  century:     Peter  Abelard  (1079-1142); 

in  the  13th  century :     Albert  the  Great  (1 193-1280) ; 
Robert  Kilwardby  (ob.  1279), 
William  of  Shyreswood  (ob.  1249), 
Peter  of  Spain  (ob  1277); 

in  the  14th  century:     William  of  Ockham  (ob.  1349/50), 

John  Buridan***  (ob.  soon  after  1358), 
Walter  Burleigh  (ob.  after  1343), 
Albert  of  Saxony  (1316-1390), 
Ralph  Strode  (ca.  1370) ; 

in  the  15th  century:     Paul  of  Venice  (ob.  1429), 

Peter    Tarteret    (wrote    between    1480    and 

Stephanus  de  Monte 

Appearance  in  this  list  comports  no  judgment  of  worth,  especially 

as  we  hardly  ever  know  whether  a  logician  was  original  or  only 

a  copyist. 

*  Verbal  communication  from  Prof.  L.  Minio-Paluello  to  whom  the  author  is 

obliged  for  much  information  about  the  12th  and  beginning  of  the  13th  century. 

**   Ockham's  productive  period  in  logic  was  wholly  prior  to  1328/29  ^25.03). 

*  *  *   Ph.  Boehner  states  that  John  of  Cornubia  (Pseudo-Scotus)  may  belong  to 

the  same  period. 



Even  the  question  of  the  literary  sources  for  Scholasticism's  new 
logical  problems  is  not  yet  satisfactorily  answered.  The  works  of 
Aristotle  provide  some  starting-points  for  the  semiotic,  especially 
the  first  five  chapters  of  the  Hermeneia  and  the  Sophistic  Refutations. 
Recent  research  shows  that  the  latter  had  a  decisive  influence  on 
the  scholastic  range  of  problems*.  But  even  the  early  scholastic 
theory  of  the  'properties  of  terms'  is  so  much  richer  and  more  many- 
sided  than  the  Aristotelian  semiotic,  that  other  influences  must  be 
supposed.  Grammar  was  certainly  an  important  one:  so  far  as  we 
can  tell,  that  was  the  basis  on  which  the  main  semiotic  problems 
were  developed  without  much  outside  influence  -  e.g.  the  whole 
doctrine  of  supposition,  the  growth  of  which  can  be  traced  with  some 

We  have  no  more  certain  knowledge  about  the  origin  of  the 
'consequences'.  Boethius's  teaching  about  hypothetical  sentences 
(rather  than  about  hypothetical  syllogisms)  was  undoubtedly  very 
influential.**  I.Thomas's  recent  inquiries  (cf.  footnote  on  30.04) 
point  to  the  Topics  as  a  principal  source;  the  Stoic  fragments  do  not 
seem  to  have  been  operative,  at  least  directly,  although  the  Outlines 
of  Pyrrhonism  of  Sextus  Empiricus  were  already  translated  into 
Latin  in  the  14th  century  (25.04).  We  do  find  doctrines  here  and 
there  which  are  recognizably  Stoic,  but  in  scholastic  logical  literature 
as  a  whole  Stoic  logic  seems  to  have  been  known  only  in  the  (obscure) 
form  of  Boethius's  syllogisms.  But  these  do  not  underlie  the  con- 
sequences, since  even  in  fairly  late  works  the  two  are  treated  in 
distinction.  Probably  scholastic  propositional  logic  is  a  rediscovery, 
starting  from  hints  in  the  Topics  and  perhaps  also  the  Hermeneia, 
rather  than  a  continuation  of  Stoic  logic. 

Arabian  logicians  certainly  exercised  some  influence,  though 
perhaps  less  than  has  commonly  been  supposed.  But  hardly  any 
research  has  been  done  on  this  subject.*** 


The  opinion  has  often  been  expressed  in  writings  on  the  history 
of  scholastic  logic  that  it  can  be  divided  firstly  according  to  schools, 
as  it  might  be  into  nominalist  and  realist  logic,  secondly  according 
to  faculties,  and  so  into  an  'artistic'  and  a  'theological'  logic.  But 
these  divisions  are  little  relevant  to   formal  logic  as  such.   More 

*  Verbal  communication  from  Prof.  Minio-Paluello. 
*  *  Prof.  E.  Moody  has  remarked  on  this  to  me. 
***    I.  Madkour's  VOrganon  d'Aristote  dans  le  monde  Arabe  (25.05)  is  quite 
inadequate.  Prof.  A.  Badawi  in  Cairo  has  published  and  discussed  a  series  of 
Arabic  logical  texts  but  unfortunately  only  in  Arabic.  Communications  received 
from  him  indicate  the  presence  of  many  interesting  doctrines. 



modern  research  has  shown  that  a  number  of  logicians  belonging 
to  sharply  opposed  philosophical  schools,  treated  of  just  the  same 
range  of  problems  and  gave  the  same  answers.  Thus  in  every  case 
we  have  met  there  is  but  one  doctrine  of  supposition,  and  differences 

are  either  to  be  ascribed  to  personal  idiosyncrasy  rather  than 
philosophical  presupposition,  or  else  are  more  epistemological  than 
logical.  Any  contrast  between  artistic  and  theological  logic  is  hardly 
more  in  place.  In  the  middle  ages  logic  was  always  part  of  the 
curriculum  of  the  faculty  of  arts,  but  no-one  was  admitted  to  the 
study  of  theology  without  having  become  Baccalaureun  artium. 
Hence  the  chief  theological  works  of  this  period  presuppose  and  use 
the  full  range  of  'artistic'  logic.  We  should  maintain  only  two 
distinctions  relevant  to  this  double  division  of  logic:  (1)  the  theolo- 
gians were  not  primarily  interested  in  logic;  (2)  some  of  them 
elaborated  logical  doctrines  of  special  importance  for  theology;  an 
example  is  the  doctrine  of  analogy  of  Thomas  Aquinas. 

Thus  in  the  Middle  Ages  we  find  essentially  only  one  logic. 
Exceptions  only  occur  where  epistemological  or  ontological  problems 
exert  an  influence,  as  in  the  determination  of  the  notion  of  logic 
itself,  and  in  the  assigning  of  denotations.  Everywhere  else  we  find  a 
unified  logic,  developing  organically.  The  very  multiplicity  of  medie- 
val views  about  extra-logical  matters  supports  the  thesis  that 
formal  logic  is  independent  of  any  special  philosophical  position  on 
the  part  of  individual  logicians. 


Our  insufficient  knowledge  of  the  period  makes  it  impossible  to 
write  a  history  of  the  evolution  of  its  logic.  A  historical  presentation 
would  be  possible  for  a  few  problems  only,  and  even  for  those  only 
for  isolated  spaces  of  time.  The  justification  of  this  chapter  in  a 
work  on  the  history  of  logical  problems  lies  in  the  fact  that,  while 
un-historical  in  itself,  it  does  to  some  extent  exhibit  one  stage  in 
the  general  development  of  logic. 

Two  questions  are  raised  by  the  choice  of  problems  for  discussion. 
The  present  state  of  research  makes  it  likely  that  we  are  not  ac- 
quainted with  them  all.  In  order  not  to  miss  at  least  the  essentials, 
we  have  made  great  use  of  the  Logica  Magna  of  Paul  of  Venice,  which 
expressly  refers  to  all  contemporary  discussions  and  may  rank  as  a 
veritable  Summa  of  14th  century  logic.  Paul's  range  of  problems 
has  been  enlarged  by  some  further  questions  from  other  authors. 

The  second  difficulty  is  posed  by  those  logical  problems  which 
overlap  epistemology  and  methodology.  Aristotle  and  the  thinkers 
of  the  Megarian-Stoic  school  envisage  them  in  a  fairly  simple  way, 
but  scholastic  conceptions  and  solutions  are  much  more  complicated. 
In  order  not  to  overstep  our  limits  too  far,  these  matters  will  be 
touched  on  only  very  superficially. 



A  survey  of  the  logical  problems  dealt  with  by  the  Scholastics 
clearly  shows  that  they  fall  into  two  classes :  on  the  one  hand  there 
are  the  ancient  ones,  Aristotelian  or  Megarian-Stoic,  concerning 
e.g.  categorical  and  modal  syllogistic,  hypothetical  syllogisms  (i.e. 
Stoic  arguments)  etc.  The  rest,  on  the  other  hand,  are  either  quite 
new,  or  else  presented  in  so  new  a  guise  as  no  longer  to  remind  one 
of  the  Greeks.  Conspicuous  in  this  class  are  the  doctrines  of  'proper- 
ties of  terms',  of  supposition,  copulation,  appellation  and  amplia- 
tion, then  too  the  doctrine  of  consequences  which  while  dependent 
on  Aristotle's  Topics  and  the  Stoics,  generalizes  the  older  teaching 
and  puts  it  in  a  new  perspective.  The  same  must  be  said  about  the 
insolubles  (§  35)  which  treat  of  the  Liar  and  such-like  but  by  new 
methods  and  in  a  much  more  general  way. 

Generally  speaking,  whatever  the  Scholastics  discuss,  even  the 
problems  of  antiquity,  is  approached  from  a  new  direction  and  by 
new  means.  This  is  more  and  more  the  case  as  the  Middle  Ages 
progress.  There  is  firstly  the  metalogical  method  of  treatment. 
Metalogical  items  are  indeed  to  be  found  in  Aristotle  (14.85 fif.), 
but  in  Scholasticism,  at  least  in  the  later  period,  there  is  nothing 
but  metalogic,  i.e.  formulae  are  not  exhibited  but  described,  so 
that  in  many  works,  e.g.  in  the  De  purilaie  artis  logicae  of  Burleigh 
not  a  single  variable  of  the  object  language  is  to  be  found.  Even 
purely  Aristotelian  matters  such  as  the  categorical  syllogism  are 
dealt  with  from  the  new  points  of  view,  semiotic  and  other.  In  early 
Scholasticism  a  double  line  of  development  is  detectable,  problems 
inherited  from  antiquity  being  treated  in  the  spirit  of  the  ancient 
logicians,  as  in  the  commentaries  of  Albert  the  Great,  and  the  new 
doctrine  being  developed  in  the  very  same  work.  Later  the  latter 
becomes  more  and  more  prominent,  so  that,  as  has  been  said,  even 
genuinely  Aristotelian  problems  are  presented  metalogically,  in 
terms  of  the  doctrine  of  supposition  etc. 

In  addition,  scholastic  logic,  even  by  the  end  of  the  13th  century, 
is  very  rich,  very  formalistic  and  exact  in  its  statement.  Some 
treatises  undoubtedly  rank  higher  than  the  Organon  and  perhaps 
than  the  Megarian-Stoic  fragments  too.  The  title  of  Burleigh's 
work  -  'De  purilaie  arlis  logicae'  -  suits  the  content,  for  here  is  a 
genuinely  pure  formal  logic. 




To  be  able  to  understand  what  the  Scholastics  thought  logic  was 
about,  one  must  be  acquainted  with  the  elements  of  their  semiotic. 
Hence  we  give  first  two  texts  from  Peter  of  Spain  followed  by  one 
from  Ockham,  about  sounds  and  terms. 


26.01  A  sound  is  whatever  is  properly  perceived  by  hearing; 
for  though  a  man  or  a  bell  may  be  heard,  this  is  only  by  means 
of  sound.  Of  sounds,  one  is  voice,  another  not  voice.  Sound- 
voice  is  the  same  as  voice;  so  voice  is  sound  produced  from 
the  mouth  of  an  animal,  formed  by  the  natural  organs.  .  .  . 
Of  voices,  one  is  literate,  another  not  literate.  Literate  voice 
is  that  which  can  be  written,  e.g.  'man';  not  literate  is  that 
which  cannot  be  written.  Of  literate  voices  one  is  significant, 
another  not  significant.  Significant  voice  is  that  which 
represents  something  to  the  hearing,  e.g.  'man'  or  the  groans 
of  the  sick  which  signify  pain.  Not  significant  voice  is  that 
which  represents  nothing  to  the  hearing,  e.g.  'bu',  'ba'.  Of 
significant  voices  one  signifies  naturally,  another  convention- 
ally. Conventionally  significant  voice  is  that  which  represents 
something  at  the  will  of  one  who  originates  it,  e.g.  'man'. 
Naturally  significant  voice  is  that  which  represents  the  same 
thing  to  all,  e.g.  the  groans  of  the  sick,  the  bark  of  dogs.  Of 
conventionally  significant  voices  one  is  simple  or  not  complex, 
e.g.  a  noun  or  a  verb,  another  composite  or  complex,  e.g.  a 
speech  (oratio).  .  .  . 

And  it  should  be  known  that  logicians  (dialedicus)  posit 
only  two  parts  of  speech,  viz.  noun  and  verb ,  calling  the  others 

26.02  Of  things  which  are  said,  some  are  said  with  com- 
plexity, e.g.  'a  man  runs',  'white  man'.  Others  without 
complexity,  e.g.  'man'  by  itself,  a  term  that  is  not  com- 
plex. ...  A  term,  as  here  understood,  is  a  voice  signifying  a 
universal  or  particular,  e.g.  'man'  or  'Socrates'. 

These  texts  contain  doctrine  generally  accepted  in  Scholasticism. 
Another,  no  less  widely  recognized,  is  excellently  summarized  by 
Ockham,  who  uses  the  expression  'terminus  conceptus'  ('conceived 
term')  instead  of  the  usual  'terminus  mentalis'  ('thought  term'). 



26.03  It  is  to  be  known  that  according  to  Boethius  .  .  . 
speech  is  threefold,  viz.  written,  spoken  and  conceived,  this 
last  having  being  only  in  the  intellect,  so  (too)  the  term  is 
threefold,  viz.  written,  spoken  and  conceived.  A  written  term 
is  part  of  a  proposition  written  down  on  some  body  which 
is  seen  or  can  be  seen  by  a  bodily  eye.  A  spoken  term  is  part 
of  a  proposition  spoken  by  the  mouth  and  apt  to  be  heard 
with  a  corporeal  ear.  A  conceived  term  is  an  intention  or 
affection  of  the  soul,  naturally  signifying  something  or  con- 
signifying,  apt  to  be  part  of  a  proposition  in  thought.  .  .  . 

Those  are  the  most  important  presuppositions  for  what  follows. 


Many  early  Scholastics  give  explicit  definitions  of  logic.  Disregard- 
ing these,  we  shall  proceed  to  descriptions  of  the  subject-matter  of 
logic,  of  which  we  know  two  kinds.  According  to  the  first  it  consists 
in  so-called  second  intentions.  Three  series  of  texts  will  illustrate  the 
matter,  taken  from  Thomas  Aquinas  (13th  century),  Ockham  and 
Albert  of  Saxiony  (early  and  late  14th  century  respectively). 

26.04  Being  is  two-fold,  being  in  thought  (ens  rationis)  and 
being  in  nature.  Being  in  thought  is  properly  said  of  those 
intentions  which  reason  produces  (adinvenit)  in  things  it 
considers,  e.g.  the  intention  of  genus,  species  and  the  like, 
which  are  not  found  among  natural  objects,  but  are  consequent 
on  reason's  consideration.  This  kind,  viz.  being  in  thought, 
is  the  proper  subject-matter  of  logic. 

26.05  The  relation  which  is  denoted  (importatur)  by  this 
name  'the  same'  is  merely  a  being  in  thought,  if  what  is  the 
same  without  qualification  is  meant:  for  such  a  relation  can 
only  consist  in  an  ordering  by  the  reason  of  something  to 
itself,  according  to  some  two  considerations  of  it. 

26.06  Because  relation  has  the  weakest  being  of  all  the 
categories,  some  have  thought  that  it  belongs  to  second 
intentions  (intelledibus).  For  the  first  things  understood  are 
the  things  outside  the  soul,  to  which  the  intellect  is  primarily 
directed,  to  understand  them.  But  those  intentions  (inten- 
tiones)  which  are  consequent  on  the  manner  of  understanding 
are  said  to  be  secondarily  understood.  ...  So  according  to 
this  thesis  (positio)  it  would  follow  that  relation  is  not  among 
the  things  outside  the  soul  but  merely  in  the  intellect,  like  the 



intention   of  genus   and   species  and  second   (i.e.   universal) 

Thus  according  to  Thomas  the  subject-matter  of  logic  is  such 
'secondarily  understood  things'  or  'second  intentions',  belonging 
to  the  domain  of  being  in  thought,  and  so  lecta.  Not  all  lecla,  however, 
but  a  special  kind,  such  as  those  corresponding  to  the  meaning  of 
logical  constants.  It  is  to  be  stressed  that  according  to  Thomas, 
as  for  the  Stoics,  the  subject-matter  of  logic  is  nothing  psychical, 
but  something  objective,  which  yet  exists  only  in  the  soul. 

The  nature  of  second  intentions  war  much  debated  among 
Scholastics,  and  we  know  of  many  different  opinions.  Ockham  says: 

26.07  It  should  first  be  known  that  that  is  called  an  'inten- 
tion of  the  soul'  which  is  something  in  the  soul  apt  to  signify 
something  else.  .  .  .  But  what  is  it  in  the  soul  which  is  such 
a  sign?  It  must  be  said  that  on  that  point  (articulum)  there 
are  various  opinions.  Some  say  that  it  is  only  something 
fashioned  by  the  soul.  Others  that  it  is  a  quality  subjectively 
existing  in  the  soul,  distinct  from  the  act  of  understanding. 
Others  say  that  it  is  the  act  of  understanding.  .  .  .  These 
opinions  will  be  examined  later.  For  the  present  it  is  enough 
to  say  that  an  intention  is  something  in  the  soul  which  is  a 
sign  naturally  signifying  something  for  which  it  can  stand 
(supponere)  or  which  can  be  part  of  a  mental  proposition. 

Such  a  sign  is  twofold.  One  which  is  a  sign  of  something 
which  is  not  such  a  sign,  .  .  .  and  that  is  called  a  'first  intention' 
such  as  is  that  intention  of  the  soul  which  is  predicable  of 
all  men,  and  similarly  the  intention  predicable  of  all  white- 
nesses, and  blacknesses,  and  so  on.  .  .  .  But  a  second  intention 
is  that  which  is  a  sign  of  such  first  intentions,  such  as  are  the 
intentions  'genus',  'species'  and  such-like.  For  as  one  intention 
common  to  all  men  is  predicated  of  all  men  when  one  says: 
'this  man  is  a  man',  'that  man  is  a  man',  and  so  on  of  each 
one,  similarly  one  intention  common  to  those  intentions 
which  signify  and  stand  for  things  is  predicated  of  them 
when  one  says:  .  .  .  'stone  is  a  species',  'animal  is  a  species, 
'colour  is  a  species'  etc. 

The  same  doctrine  is  further  developed  by  Albert  of  Saxony: 

26.08  'Term  of  first  intention'  is  the  name  given  to  that 
mental  term  which  is  significative  of  things  not  from  the 
point  of  view  of  their  being  signs.  Thus  this  mental  term 
'man',  or  this  mental  term  'being',  or  this  mental  term  'qua- 



lity',  or  this  mental  term  'voice'.  Hence  this  mental  term 
'man'  signifies  Socrates  or  Plato,  and  not  insofar  as  Socrates 
or  Plato  are  signs  for  other  things.  .  .  .  But  a  mental  term 
which  is  naturally  significative  of  things  insofar  as  they  are 
signs  is  called  a  'term  of  second  intention',  and  if  they  ceased 
to  be  signs  it  would  not  signify  them.  Of  this  kind  are  the 
mental  terms  'genus',  'species',  'noun',  'verb',  'case  of  a  noun' 

In  the  last  two  texts  the  conception  is  other  than  that  of  Thomas. 
Second  intentions  are  there  conceived  in  a  purely  semantic  way; 
they  are  signs  of  signs,  and  for  Albert  signs  of  signs  as  such. 

Whether  Ockham  and  Albert  thought  of  logic  as  in  any  sense  a 
science  of  second  intentions  remains  open  to  question.  One  might 
perhaps  give  expression  to  both  their  views  by  saying  that  logic 
is  a  science  constructed  throughout  in  a  meta-language,*  remarking 
at  the  same  time  that  the  Scholastics  included  under  'signs'  mental 
as  well  as  exterior  (written  or  spoken)  signs. 

However,  one  common  feature  underlies  all  these  fundamental 
differences;  logic  is  sharply  distinguished  from  ontology  in  the 
whole  scholastic  tradition.  This  is  so  for  Thomas,  since  its  object 
is  not  real  things,  but  second  intentions;  and  for  his  successors,  since 
it  is  expressed  not  in  an  object-  but  in  a  meta-language. 

It  should  also  be  noted  that  in  fact  the  entire  practice  of  medieval 
logic  corresponds  to  the  Thomist  conception  of  the  object  of  logic, 
even  though  this  conception  was  not  the  only  one.  For  scholastic 
logic  essentially  consists  of  two  parts:  the  doctrine  of  the  properties 
of  terms,  and  the  doctrine  of  consequences.  The  properties  of  terms 
are  evidently  second  intentions  in  the  Thomist  sense;  and  one  must 
think  oi  consequences  in  the  same  light,  since  the  logical  relationships 
they  exhibit  (e.g.  between  antecedent  and  consequent)  are  not  real 


There  is  a  difficulty  in  adopting  the  view  that  we  have  hypotheti- 
cally  ascribed  to  Ockham  about  the  subject  matter  of  logic,  in  that 
it  does  not  achieve  a  definition  of  logic  as  a  distinct  science,  since 
every  science  can  be  formulated  in  a  meta-language.  But  we  find, 
though  not  explicitly,  logic  limited  to  concern  with  logical  form, 
which  leads  to  an  exact  definition  of  formal  logic  when  this  form 
is  equated  with  the  syncategoremata.  Scholastic  practice  is  wholly  in 
accord  with  this  definition  in  its  cultivation  of  the  corresponding 

*  I  am  particularly  obliged  to  Prof.  E.  Moody  for  valuable  assistance  with 
these  questions. 



theory  of  logical  form.  Three  texts  about  Byncategoremata  follow, 
one  from  William  of  Shyreswood  (13th  century),  one  from  Ockham 
(beginning  of  14th)  and  one  from  Buridan*  (end  of  14th). 

26.09  To  understand  propositions  one  must  know  their 
parts.  Their  parts  are  twofold,  primary  and  secondary. 
Primary  parts  are  substantival  names  and  verbs;  these  are 
necessary  for  an  understanding  of  propositions.  Secondary 
parts  are  adjectival  names,  adverbs,  conjunctions  and  prepo- 
sitions; these  are  not  essential  to  the  constitution  of  proposi- 

Some  secondary  parts  are  determinations  of  primary  ones 
with  reference  to  (ratione)  their  things  (i.e.  to  which  they 
refer),  and  such  are  not  syncategoremata ;  e.g.  when  I  say 
'white  man'  'white'  signifies  that  one  of  its  things,  a  man,  is 
white.  Others  are  determinations  of  primary  parts  insofar  as 
these  are  subjects  or  predicates;  e.g.  when  I  say  'every  man 
runs',  the  'every',  which  is  a  universal  sign,  does  not  mean 
that  one  of  its  things,  namely  a  man,  is  universal,  but  that 
'man'  is  a  universal  subject.  Such  are  called  'syncategoremata' 
and  will  be  treated  (here),  as  offering  considerable  difficulties 
in  discourse. 

Ockham  affords  a  development  of  the  same  thought: 

26.10  Categorematic  terms  have  a  definite  and  certain 
signification,  e.g.  this  name  'man'  signifies  all  men,  and  this 
name  'animal'  all  animals,  and  this  name  'whiteness'  all 
whitenesses.  But  syncategorematic  terms,  such  as  are  'all', 
'no',  'some',  'whole',  'besides',  'only',  'insofar  as'  and  such- 
like, do  not  have  a  definite  and  certain  signification,  nor  do 
they  signify  anything  distinct  from  what  is  signified  by  the 
categoremata.  Rather,  just  as  in  arabic  numeration  a  zero 
(cifra)  by  itself  signifies  nothing,  but  attached  to  another 
figure  makes  that  signify,  so  a  syncategorema  properly  speak- 
ing signifies  nothing,  but  when  attached  to  something  else 
makes  that  signify  something  or  stand  for  some  one  or  more 
things  in  a  determinate  way,  or  exercises  some  other  function 
about  a  categorema.  Hence  this  syncategorema  'all'  has  no 
definite   significate,   but  when  attached   to   'man'   makes   it 

*  It  is  taken  from  the  Consequentiae  which  is  ascribed  to  Buridan  in  the  early 
printed  editions,  though  a  letter  from  Fr.  Ph.  Boehner  informs  us  that  no  MS  of 
this  work  has  yet  been  found. 



stand  or  suppose  for  all  men  .  .  .  ,  and  attached  to  'stone' 
makes  it  stand  for  all  stones,  and  attached  to  'whiteness' 
makes  it  stand  for  all  whitenesses.  And  the  same  is  to  be  held 
proportionately  for  the  others,  as  for  that  syncategorema 
'all',  though  distinct  functions  are  exercised  by  distinct 
syncategoremata,  as  will  later  be  shown  for  some  of  them. 

Evidently,  the  syncategoremata  are  our  logical  constants.  That 
they  determine  logical  form  is  expressly  and  consciously  propound- 
ed by  Buridan  (whose  text  was  later  adopted  almost  word  for 
word  by  Albert  of  Saxony:  26.11). 

26.12  When  form  and  matter  are  here  spoken  of,  by  the 
matter  of  a  proposition  or  consequence  is  understood  merely 
the  categorematic  terms,  i.e.  the  subject  and  predicate,  to 
the  exclusion  of  the  syncategorematic  *  ones  attached  to 
them,  by  which  they  are  restricted,  negated,  or  divided  and 
given  (trahuntur)  a  determinate  kind  of  supposition.  All  else, 
we  say,  belongs  to  the  form.  Hence  we  say  that  the  copula, 
both  of  the  categorical  and  of  the  hypothetical  proposition 
belongs  to  the  form  of  the  proposition,  as  also  negations, 
signs,  the  number  both  of  propositions  and  terms,  as  well 
as  the  mutual  ordering  of  all  the  aforesaid,  and  the  inter- 
connections of  relative  terms  and  the  ways  of  signifying 
(modos  significandi)  which  relate  to  the  quantity  of  the 
proposition,  such  as  discreteness**,  universality  etc.  .  .  . 

E.g.  .  .  Since  modals  have  subordinate  copulas  and  so  differ 
from  assertoric  propositions,  these  differ  in  form ;  and  by  reason 
of  the  negations  and  signs  (signa)  affirmatives  are  of  another 
form  than  negatives,  and  universals  than  particulars;  and  by 
reason  of  the  universality  and  discreteness***  of  their  terms 
singular  propositions  are  of  another  form  than  indefinites ;  by 
reason  of  the  number  of  terms  the  following  propositions  are 
of  different  forms:  'man  is  man'  and  'man  is  ass',  as  are  the 
following  consequences  or  hypothetical  propositions:  'every 
man  runs,  therefore  some  man  runs'  and  'every  man  runs, 
therefore  some  ass  walks  about'.  Similarly  by  reason  of  the 
order  the  following  are  of  different  forms:  'every  man  is 
animal',  'animal  is  every  man',  and  likewise  the   following 

*   Reading  syncategoremaiicis  for  categoricis. 
*  *   Reading  discretio  for  descriptio. 
*  *  *  See  last  note. 



consequences:  'every  B  is  A,  therefore  some  B  is  A1  and 
'every  B  is  A,  therefore  some  A  is  B'  etc.  Similarly  by  rea- 
son of  the  relationship  and  connection  .  .  .  'the  man  runs,  the 
man  does  not  run'  is  of  another  form  than  this:  'the  man  runs 
and  the  same  does  not  run':  since  its  form  makes  the  second 
impossible,  but  it  is  not  so  with  the  first. 

It  is  easy  to  establish  that  scholastic  logic  has  for  its  object 
precisely  form  so  conceived.  The  doctrine  of  the  properties  of  terms 
treats  of  supposition,  appellation,  ampliation  and  such-like  rela- 
tionships, all  of  which  are  determined  in  the  proposition  by  syn- 
categorematic  terms;  while  the  second  part  of  scholastic  logic, 
comprising  the  doctrine  of  the  syllogism,  consequences  etc.,  treats 
of  formal  consequence,  which  holds  in  virtue  of  the  form  as  de- 

Expressed  in  modern  terms,  the  difference  between  the  two 
conceptions  of  logic  that  have  been  exemplified,  is  that  the  first  is 
semantic,  the  second  syntactical:  for  the  first  uses  the  idea  of 
reference,  the  second  determines  logical  form  in  a  purely  structural 
way.  According  to  the  second  the  logical  constants  are  the  subject- 
matter  of  logic,  while  on  the  view  of  Thomas  this  object  is  their 
sense.  On  either  view  Scholasticism  achieved  a  very  clear  idea  of 
logical  form  and  so  of  logic  itself. 


Two  kinds  of  logical  works  can  be  distinguished  in  Scholasticism, 
commentaries  on  Aristotle  and  independent  treatises  or  manuals. 
To  begin  with,  the  composition  of  works  even  of  the  second  kind 
is  strongly  influenced  by  the  Aristotelian  range  of  problems,  at  least 
in  the  sense  that  newer  problems  are  incorporated  into  the  frame- 
work of  the  Organon.  It  is  only  gradually  that  the  ever  growing 
importance  of  the  new  problems  finds  expression  in  the  very  con- 
struction of  the  works.  We  shall  show  this  in  some  examples  collected 
for  the  most  part  by  Ph.  Boehner  (26.13). 

Albert  the  Great  has  no  independent  arrangement;  his  logic 
consists  of  commentaries  on  the  writings  of  Aristotle  and  Boethius. 

The  chief  logical  work  of  Peter  of  Spain  falls  into  two  parts;  the 
first  is  markedly  Aristotelian  and  contains  the  following  treatises: 
On  Propositions  (=  Hermeneia), 
On  the  Predicables  (=  Porphyry), 
On  the  Categories  (=  Categories), 
On  Syllogisms  (=  Prior  Analytics), 
On  Loci  (=  Topics), 
On  Fallacies  (=  Sophistic  Refutations). 



In  the  second  part  is  to  be  found  nothing  but  the  new  problematic, 
for  it  is  divided  into  treatises  on 






Two  points  are  notable:  that  propositions  are  discussed  at  the  start 
(and  not  in  the  third  place  as  in  Porphyry  and  the  Categories),  and 
that  the  doctrine  of  supposition  is  inserted  before  the  treatise  on 
fallacies.  That  shows  how  the  new  problematic  began  to  influence 
the  older  one. 

Ockham's  Summa  is  divided  in  another  way: 
I.  Terms: 

1.  In  general. 

2.  Predicables. 

3.  Categories. 

4.  Supposition. 

II.  Propositions: 

1.  Categorical  and  modal  propositions. 

2.  Conversion. 

3.  Hypothetical  propositions. 

III.  Arguments: 

1.  Syllogisms: 

a)  assertoric, 

b)  modal, 

c)  mixed  (from  the  first  two  kinds), 

d)  'exponibilia', 

e)  hypothetical. 

2.  Demonstration  (in  the  sense  of  the  Posterior  Analytics). 

3.  Further  rules: 

a)  Consequences. 

b)  Topics. 

c)  Obligations. 

d)  Insolubles. 

4.  Sophistics. 

The  general  framework  here  is  still  Aristotelian,  more  so  even  than 
with  Peter,  but  the  new  problematic  has  penetrated  into  the 
subdivisions.  An  Aristotelian  title  often  conceals  strange  material, 
as  when  the  chapter  on  the  categories  deals  with  typically  scholastic 
problems  about  intentions  etc. 



Walter  Burleigh's  De  purilale  artis  logicae  is  divided  thus: 
I.  On  terms: 

1.  Supposition. 

2.  Appellation. 

3.  Copulation. 

II.  (Without  title): 

1.  Hypothetical  propositions. 

2.  Conditional  syllogisms. 

3.  Other  hypothetical  syllogisms. 

Even  this  small  sample  shows  how  the  scholastic  range  of  problems 
is  to  the  fore. 

Albert  of  Saxony  divides  his  logic  in  this  way: 

1.  Terms  (in  general). 

2.  Properties  of  terms   (supposition,  ampliation,  appellation). 

3.  Propositions. 

4.  Consequences: 

a)  in  general. 

b)  Propositional  consequences. 

c)  Syllogistic  consequences. 

d)  Hypothetical  syllogisms. 

e)  Modal  syllogisms. 

f)  Topics. 

5.  Sophistics. 

6.  Antinomies  and  obligations. 

Here  the  whole  of  Aristotelian  and  Stoic  formal  logic  has  been  built 
into  the  scholastic  doctrine  of  consequences,  while  this  last  is 
introduced  by  discussion  of  another  typically  scholastic  matter, 
the  properties  of  terms. 

Finally  we  consider  the  division  of  the  Logica  Magna  of  Paul 
Nicollet  of  Venice  (ob.  1429),  which  is  probably  the  greatest  syste- 
matic work  on  formal  logic  produced  in  the  Middle  Ages.  It  falls 
into  two  parts,  the  first  designed  to  treat  of  terms,  the  second  of 
propositions,  though  in  fact  the  first  contains  much  about  propo- 
sitions, and  the  second  includes  also  the  doctrine  of  consequences 
and  syllogisms. 

Part  I : 

1.  Terms. 

2.  Supposition. 

3.  Particles  that  cause  difficulty. 

4.  Exclusive  particles. 

5.  Rules  of  exclusive  propositions. 

6.  Exceptive  particles. 

7.  Rules  of  exceptive  propositions. 

8.  Adversative  particles. 



9.  'How'. 

10.  Comparatives. 

11.  Superlatives. 

12.  Objections  and  counter-arguments. 

13.  Categorematic  'whole'  (totus). 

14.  'Always'  and  'ever'. 

15.  'Infinite'. 

16.  'Immediate'. 

17.  'Begins'  and  'ceases'. 

18.  Exponible  propositions. 

19.  Propositio  officiabilis. 

20.  Composite  and  divided  sense. 

21.  Knowing  and  doubting. 

22.  Necessity  and  contingence  of  future  events. 

Part  II : 

1.  Propositions  (in  general). 
2.-3.  Categorical  propositions. 

4.  Quantity  of  propositions. 

5.  Logical  square. 

6.  Equivalences. 

7.  Nature  of  the  proposition  in  the  square. 

8.  Conversion. 

9.  Hypothetical  propositions. 

10.  Truth  and  falsity  of  propositions. 

11.  Signification  of  propositions. 

12.  Possibility,  impossibility. 

13.  Syllogisms. 

14.  Obligations. 

15.  Insolubles. 

Here  the  treatise  on  consequences  has  disappeared,  having  been 
incorporated  into  that  on  hypothetical  propositions.* 


We  begin  our  presentation  of  scholastic  logic  with  the  doctrine 
of  supposition.  This  is  one  of  the  most  original  creations  of  Schola- 
sticism,   unknown   to    ancient   and   modern   logic,    but   playing   a 

*  The  following  figures  will  give  an  idea  of  the  scope  of  this  work.  The  Logica 
Magna  occupies  199  folios  of  four  columns  each  containing  some  4600  printed 
signs,  so  that  the  whole  work  comprises  about  3,650,000  signs.  This  corresponds 
to  at  least  1660  normal  octavo  pages,  four  to  five  volumes.  But  the  Logica  Magna 
is  only  one  of  four  works  by  Paul  on  formal  logic,  the  others  together  being  even 
more  voluminous.  None  of  it  is  merely  literary  work,  but  a  pure  logic,  written  in 
terse  and  economical  language. 



central  role  here.  Unpublished  research  of  L.  Minio  Paluello  enables 

us  to  trace  its  origin  to  the  second  hall  of  the  12th  century.  By  the 
middle  of  the  13th  all  available  sources  witness  to  its  being  every- 
where accepted.  Later  there  appear  some  developments  of  detail, 
but  no  essentially  new  fundamental  ideas. 

We  shall  first  illustrate  the  notion  of  supposition  in  general,  then 
proceed  to  the  theory  of  material  and  simple  supposition,  and  finally 
mention  other  kinds. 


The  notion  of  supposition  is  already  well  defined  in  Shyreswood, 
and  distinguished  by  him  from  similar  'properties  of  terms' : 

27.01  Terms  have  four  properties,  which  we  shall  now 
distinguish.  .  .  .  These  properties  are  signification,  supposition, 
copulation  and  appellation.  Signification  is  the  presentation 
of  a  form  to  the  reason.  Supposition  is  the  ordering  of  one 
concept  (intelledus)  under  another.  Copulation  is  the  ordering 
of  one  concept  over  another.  It  is  to  be  noted  that  supposition 
and  copulation,  like  many  words  of  this  kind,  are  proffered 
(dicuntur)  in  two  senses,  according  as  they  are  supposed  to  be 
actual  or  habitual.  Their  definitions  belong  to  them  according 
as  they  are  supposed  to  be  actual.  But  insofar  as  they  are 
supposed  to  be  habitual,  'supposition'  is  the  name  given 
to  the  signification  of  something  as  subsisting;  for  what 
subsists  is  naturally  apt  to  be  ordered  under  another.  And 
'copulation'  is  the  name  given  to  the  signification  of  some- 
thing as  adjacent,  for  what  is  adjacent  is  naturally  apt  to  be 
ordered  over  another.  But  appellation  is  the  present  attribution 
of  a  term,  i.e.  the  property  by  which  what  a  term  signifies  can 
be  predicated  of  something  by  means  of  the  verb  'is'. 

It  follows  that  signification  is  present  in  every  part  of 
speech,  supposition  only  in  substantives,  pronouns  or  sub- 
stantival particles;  for  these  (alone)  signify  the  thing  as 
subsistent  and  of  such  a  kind  as  to  be  able  to  be  set  in  order 
under  another.  Copulation  is  in  all  adjectives,  participles  and 
verbs,  appellation  in  all  substantives,  adjectives  and  parti- 
ciples, but  not  in  pronouns  since  these  signify  substance  only, 
not  form.  Nor  is  it  in  verbs.  .  .  .  None  of  these  three,  supposi- 
tion, copulation  and  appellation  is  present  in  the  indeclinable 
parts  (of  speech),  since  no  indeclinable  part  signifies  substance 
or  anything  in  substance. 


Thomas  Aquinas  speaks  in  similar  fashion: 

27.02  The  proper  sense  (ratio)  of  a  name  is  the  one  which 
the  name  signifies;  .  .  .  But  that  to  which  the  name  is  attri- 
buted if  it  be  taken  directly  under  the  thing  signified  by  the 
name,  as  determinate  under  indeterminate,  is  said  to  be 
supposed  by  the  name;  but  if  it  be  not  directly  taken  under 
the  thing  of  the  name,  it  is  said  to  be  copulated  by  the  name; 
as  this  name  'animal'  signifies  sensible  animate  substance, 
and  'white'  signifies  colour  disruptive  of  sight,  while  'man' 
is  taken  directly  under  the  sense  of  'animal'  as  determinate 
under  indeterminate.  For  man  is  sensible  animate  substance 
with  a  particular  kind  of  soul,  viz.  a  rational  one.  But  it  is 
not  directly  taken  under  white,  which  is  extrinsic  to  its  essence. 

27.03  The  difference  between  substantives  and  adjectives 
consists  in  this,  that  substantives  refer  to  (ferunt)  their 
suppositum,  adjectives  do  not,  but  posit  in  the  substance* 
that  which  they  signify.  Hence  the  logicians  (sophistae)  say 
that  substantives  suppose,  adjectives  do  not  suppose  but 

The  doctrine  implicit  in  these  texts  was  later  expressly  formulated 
by  Ockham: 

27.04  (Supposition)  is  a  property  belonging  to  terms,  but 
only  as  (they  occur)  in  a  proposition. 


Shyreswood  writes: 

27.05  Supposition  is  sometimes  material,  sometimes  formal. 
It  is  called  material  when  an  expression  (diciio)  stands  either 
for  an  utterance  (vox)  by  itself,  or  for  the  expression  which 
is  composed  of  an  utterance  and  (its)  significance,  e.g.  if 
we  were  to  say:  'homo'  consists  of  two  syllables,  'homo'  is 
a  name.  It  is  formal  when  an  expression  stands  for  what  it 

27.06  The  first  division  of  supposition  is  disputed.  For  it 
seems  that  kinds  not  of  supposition  but  of  signification  are 
there  distinguished.  For  signification  is  the  presentation  of  a 
form  to  the  reason.  So  that  where  there  is  different  presenta- 

Reading  substantiam  for  substantivum. 



tion  there  is  different  signification.  Now  when  an  expression 
supposes  materially  it  presents  either  itself  or  its  utterance; 
but  when  formally,  it  presents  what  it  signifies;  therefore  it 
presents  something  different  (in  each  case);  therefore  it 
signifies  something  different.  But  that  is  not  true,  since 
expressions  by  themselves  always  present  what  they  signify, 
and  if  they  present  their  utterance  they  do  not  do  this  of 
themselves  but  through  being  combined  with  a  predicate.  For 
some  predicates  naturally  refer  to  the  mere  utterance  or  to 
the  expression,  while  others  refer  to  what  is  signified.  But  this 
effects  no  difference  in  the  signification.  For  the  expression  as 
such,  before  ever  being  incorporated  in  a  sentence,  already 
has  a  significance  which  does  not  arise  from  its  being  co- 
ordinated with  another. 

On  this  question  Thomas  Aquinas  remarks: 

27.07  One  could  object  to  this  (teaching  of  ours)  also, 
that  verbs  in  other  moods  (than  the  infinitive)  seem  to  be  put 
as  subjects,  e.g.  if  one  says:  'I  run  is  a  verb'.  But  it  must  be 
said  that  the  verb  'I  run'  is  not  taken  formally  in  this  state- 
ment (locutio).  (i.e.)  with  its  signification  referred  to  a  thing, 
but  as  materially  signifying  the  word  itself  which  is  taken  as  a 

The  expressions  'suppositio  materialis'  and  'suppositio  formalis' 
have  also  another  meaning  for  Thomas.  He  sometimes  uses  the 
first  for  suppositio  personalis  (cf.  27.23  ff.)  and  the  second  for 
suppositio  simplex  (cf.  27.17f.) : 

27.08  A  term  put  as  subject  holds  (tenetur)  materially, 
i.e.  (stands)  for  the  suppositum;  but  put  as  predicate  it 
holds  formally,  i.e.  (stands)  for  the  nature  signified. 

Perhaps  this  ambiguity  accounts  for  the  expression  'formal 
supposition',  that  we  have  found  in  Shyreswood  and  Thomas,  later 
disappearing,  so  far  as  we  know,  outside  the  Thomist  school.* 
Even  by  Ockham's  time  supposition  is  divided  immediately  into 
three  kinds: 

27.09  Supposition  is  first  divided  into  personal,  simple 
and  material. 

*Fr.  Ph.  Boehner  is  to  be  thanked  for  the  information  that  this  expression 
occurs  in  Chr.  Javellus  (ob.  1538). 



The  two  first  of  those  are  sub-species  of  the  formal  supposition  of 
Shyreswood  and  Thomas,  which  Ockham  no  longer  refers  to.  His 
division  is  subsequently  the  usual  one,  except  among  the  Thomists. 
In  27.05  we  read  of  an  'utterance  by  itself  and  an  'expression 
which  is  composed  of  an  utterance  and  (its)  significance'.  This 
distinction  is  developed  at  the  end  of  the  15th  century  by  Peter 

27.10  Material  supposition  is  the  acceptance  of  a  term  for 

its  non-ultimate  significate,  or  its  non-  ultimate  significates 

In  which  it  is  to  be  noticed  that  significates  are  two-fold, 
ultimate  and  non-ultimate.  The  ultimate  significate  is  that 
which  is  ultimately  signified  by  a  term  signifying  conven- 
tionally, and  ultimately  or  naturally  and  properly.  But  the 
non-ultimate  significate  is  the  term  itself,  or  one  vocally  or 
graphically  similar,  or  one  mentally  equivalent.  From  which 
it  follows  that  a  vocal  or  written  term  is  said  to  signify 
conventionally  in  two  ways,  either  ultimately  or  non-ulti- 
mately.  Ultimately  it  signifies  what  it  is  set  to  signify;  but 
a  vocal  term  is  said  to  signify  conventionally  and  non-ulti- 
mately  a  synonymous  written  term;  and  a  written  term 
is  said  to  signify  non-ultimately  an  utterance  synonymous 
with  it.  .  .  . 

From  the  modern  point  of  view  this  doctrine  reflects  our  distinc- 
tion of  language  and  meta-language,  except  that  in  place  of  two 
languages,  symbols  of  one  language  exercise  a  two-fold  supposition. 
Furthermore,  the  two  last-cited  texts  exhibit  the  important  distinc- 
tion between  the  name  of  an  individual  symbol  and  the  name  of  a 
class  of  equiform  symbols.  We  do  not  find  this  in  the  logistic  period 
till  after  1940. 

This  distinction  first  occurs,  so  far  as  we  know,  in  St.  Vincent 
Ferrer*  (14th  century),  as  a  division  of  material  supposition: 

27.11  Material  supposition  is  divided  as  is  formal.  One 
(kind  of  material  supposition)  is  common,  the  other  discrete. 

*  Vincent  Ferrer  was  the  greatest  preacher  of  his  time.  We  would  add  that 
Savonarola  was  also  an  important  logician.  A  similar  link  between  deep  religiDUS 
life  and  a  talented  interest  in  formal  logic  is  also  to  be  observed  in  Indian  culture 
especially  among  the  Buddhists.  This  would  seem  to  be  a  little  known  and  as  yet 
unexplained  phenomenon.  The  authenticity  of  Vincent  Ferrer's  philosophical 
opuscules  De  Suppositionibus  and  De  imitate  universalis  has  been  challenged  so 
far  as  we  know  only  by  S.  Brettle  (vid.  Additions  to  Bibliography  3.98).  To  his 
p.  105  note  3  should  be  added  a,  here  relevant,  reference  to  p.  33  note  10.  M.  G. 
Miralles  (vid.  Additions)  summarizes  the  arguments  for  and  against,  and 
concludes  with  M.  Gorce  (vid.  Additions):  'L'authenticite  des  deux  6crits  n'a  ete 
jamais  mise  en  doute.  Le  temoignage  du  contemporain  Ranzzano  suffit  a  la 



It  is  discrete  if  the  term  or  utterance  stands  determinately 
for  a  suppositum  of  its  material  significate.  And  thus  discrete 
material*  suppositum  occurs  in  three  ways.  In  one  way 
through  the  utterance  or  term  itself,  as  when  one  asks: 
'What  is  it  you  want  to  say?'  and  the  other  answers:  'I  say 
"buf"  and  '"baf"  is  said  by  me',  (then)  the  subject  of  this 
proposition  supposes  materially  and  discretely  since  it  stands 
for  the  very  utterance  numerically  identical  (with  it)  (cf. 
11.11).  This  becomes  more  evident  if  names  are  assigned  to 
the  individual  terms  in  such  a  way  that  as  this  name  'man'  * 
signifies  this  individual  man  so  this  name  lA'  signifies  that 
individual  word  'buf'  and  '£'  the  other  ('baf').  And  then  if 
it  is  said:  lA  is  an  utterance'  or  'A  is  said  by  me',  the 
subject  supposes  materially  and  discretely,  as  in  the  proposi- 
tion 'Socrates  runs'  the  subject  supposes  formally  and  dis- 

27.12  It  occurs  secondly  through  a  demonstrative  name 
(nomen)  demonstrating  an  utterance  or  singular  term,  as 
when  the  utterance  of  the  term  'man'  is  written  somewhere 
and  one  says,  with  reference  to  this  utterance:  'That  is  a 
name'.  Then  the  subject  of  the  proposition  supposes  materially 
for  that  which  it  demonstrates. 

It  occurs  in  a  third  way  through  a  term.  .  .  .,  which  is 
determined  by  a  demonstrative  pronoun,  as  when  it  is  said 
of  the  written  utterance  'man':  'this  "man"  is  a  name'  or 
'this  utterance  is  a  name'. 

And  each  of  these  ways  .  .  .  can  be  varied  by  natural, 
personal  or  simple  supposition,  as  was  said  about  singular 
formal  supposition. 

Common  (communis)  material  supposition  is  when  the 
utterance  or  term  stands  indeterminately  for  its  material 
signification,  as  when  it  is  said:  "people"  is  written'  the  sub- 
ject of  this  proposition  stands  indeterminately  for  this  term 
'people',  or  (in  another  example)  for  some  other  (term).  I 
do  not  say  that  in  the  proposition  '"people"  is  written'  or  in 
some  other  such  that  the  supposition  is  indeterminate,  but 
that  the  subject  is  indeterminate  and  is  taken  indeterminately 

Material  common  supposition  is  divided  into  natural, 
personal  and  simple  supposition,  like  common  formal  supposi- 

*   Reading  materialis  for  formalis. 
**   Reading  homo  for  primo. 



tion.  . . .  An  example  of  personal :  '  "man"  is  heard', '  "man"  is 
written ','  "man"  is  answered'.  An  example  of  simple 
(supposition):  '"man"  is  a  species  of  utterance',  '"man"  is 
conceived',  '"man"  is  said  by  this  man';  and  so  on  in  many 
other  cases  as  everyone  can  see  for  himself. 

So  material  supposition  is  divided  just  as  is  formal.  These  texts 
exhibit  scholastic  semantics  at  its  best.  This  accuracy  of  analysis 
is  the  more  astonishing  when  one  remembers  that  the  distinction 
mentioned  in  the  introduction  to  27.11  remained  unknown,  not 
only  to  the  decadent  'classical'  logicians,  but  also  to  mathematical 
ones  for  nearly  a  century. 

It  should  also  be  noticed  that  in  the  text  of  Tarteret  just  cited,  a 
distinction  occurs  which  cannot  be  expressed  in  contemporary 
terms.  The  Scholastics  distinguished,  as  has  already  be  said  above 
(26.03)  three  inter-related  kinds  of  sign:  graphical,  vocal  and 
psychic,  and  a  materially  supposing  graphical  sign  can  stand  either 
for  itself  (or  its  equiforms)  or  for  the  corresponding  vocal  or  psychic 

Burleigh  has  another  division,  parallel  to  that  between  material 
and  formal  supposition: 

27.13  The  tenth  rule  is:  that  on  every  act  that  is  accom- 
plished there  follows  the  act  that  is  signified,  and  conversely. 
For  it  follows:  'man  is  an  animal,  therefore  "animal"  is 
predicated  of  "man"',  for  the  verb  'is'  accomplishes  predica- 
tion, and  this  verb  'is  predicated'  signifies  predication,  and 
syncategorematic  particles  accomplish  acts,  and  adjectival 
verbs  signify  such  acts.  E.g.  the  sign  'all'  accomplishes 
distribution,  and  the  verb  'to  distribute'  signifies  distributions; 
the  particle  'if  exercises  consequence,  and  this  verb  'it  follows' 
signifies  consequence. 

It  was  said  above  that  this  distinction  runs  parallel  with  that 
between  formal  and  material  supposition,  for  it  could  easily  be 
translated  into  it.  But  Burleigh  would  not  seem  to  be  thinking  of 
these  suppositions  here;  by  'the  act  signified'  he  means  not  words, 
but  their  significates.  For  in  his  example,  the  word  'animal'  is  not 
predicated  of  the  word  'man',  but  what  the  first  signifies  is  predi- 
cated of  that  for  which  the  second  supposes. 


Along  with  the  idea  of  material  supposition,  that  of  simple 
(simplex)  supposition  is  an  interesting  scholastic  novelty.  On  this 
subject  we  can  limit  ourselves  to  the  13th  century,  and  mainly  to 



Peter  of  Spain.  First  we  shall  give  some  of  his  general  divisions  of 
formal  supposition : 

27.14  One  kind  of  supposition  is  common,  another  discrete. 
Common  supposition  is  effected  by  a  common  term  such  as 
'man'.  Again  of  common  suppositions  one  kind  is  natural, 
another  accidental.  Natural  supposition  is  the  taking  of  a 
common  term  for  everything  of  which  it  is  naturally  apt  to  be 
predicated,  as  'man'  taken  by  itself  naturally  possesses 
supposition  for  all  men  who  are  and  who  have  been  and  who 
will  be.  Accidental  supposition  is  the  taking  of  a  common 
term  for  everything  for  which  its  adjunct  requires  (it  to  be 
taken).  E.g.  'A  man  exists';  the  term  'man'  here  supposes  for 
present  men.  But  when  it  is  said:  'a  man  was',  it  supposes  for 
past  men.  And  when  it  is  said :  'a  man  will  be',  it  supposes  for 
future  ones,  and  so  has  different  suppositions  according  to  the 
diversity  of  its  adjuncts. 

Later  on  we  also  meet  with  an  'improper'  (27.15)  and  a  'mixed' 
(27.16)  supposition.  The  first  simply  consists  in  the  metaphorical 
use  of  a  term.  The  second  was  introduced  to  elucidate  the  function 
of  terms  of  which  one  part  supposed  in  one  way,  another  in  another. 
From  the  logical  point  of  view  these  are  not  very  important  ideas. 
Of  greater  importance  is  Peter's  continuation : 

27.17  Of  accidental  suppositions  one  is  simple,  another 
personal.  Simple  supposition  is  the  taking  of  a  common  term 
for  the  universal  thing  symbolized  (figurata)  by  it,  as  when  it 
is  said :  'man  is  a  species'  or  'animal  is  a  genus',  the  term  'man' 
supposes  for  man  in  general  and  not  for  any  of  its  inferiors, 
and  similarly  in  the  case  of  any  common  term,  as  'risible  is  a 
proprium',  'rational  is  a  difference'. 

27.18  Of  simple  suppositions  one  belongs  to  a  common 
term  set  as  subject,  as  'man  is  a  species';  another  belongs  to  a 
common  term  set  as  an  affirmative  predicate,  as  'every  man 
is  an  animal';  the  term  'animal'  set  as  a  predicate  has  simple 
supposition  because  it  only  supposes  for  the  generic  nature; 
yet  another  belongs  to  a  common  term  put  after  an  exceptive 
form  of  speech,  as  'every  animal  apart  from  man  is  irrational'. 
The  term  'man'  has  simple  supposition.  Hence  it  does  not 
follow:  'every  animal  apart  from  man  is  irrational,  therefore 
every  animal  apart  from  this  man  (is  irrational)',  for  there 
is  there  the  fallacy  of  the  form  of  speech  (cf.  11.19),  when 
passage  is  made  from  simple  to  personal  supposition.  Similarly 



here:  'man  is  a  species,  therefore  some  man  (is  a  species)'.  In 
all  such  cases  passage  is  made  from  simple  to  personal  sup- 

27.19  But  that  a  common  term  put  as  predicate  is  to  be 
taken  with  simple  supposition  is  clear  when  it  is  said :  'of  all 
contraries  there  is  one  and  the  same  science',  for  unless  the 
term  'science'  had  simple  supposition  there  would  be  a  fallacy. 
For  no  particular  science  is  concerned  with  all  contraries; 
medicine  is  not  concerned  with  all  contraries  but  only  with 
what  is  healthy  and  what  is  sick,  and  grammar  with  what  is 
congruous  and  incongruous,  and  so  on. 

This  is  to  be  compared  with  the  text  of  Thomas  cited  above 
(27.08).  The  following  text  from  him  from  a  theological  context 
expresses  the  matter  clearly: 

27.20  The  proposition  homo  fadus  est  Deus  .  .  .  can  be 
understood  as  though  fadus  determines  the  composition,  so 
that  the  sense  would  be:  'a  man  is  in  fact  God',  i.e.  it  is  a  fact 
that  a  man  is  God.  And  in  this  sense  both  are  true,  homo  fadus 
est  Deus  and  Deus  fadus  est  homo.  But  this  is  not  the  proper 
sense  of  these  propositions  (locutionum),  unless  they  were  to  be 
so  understood  that  'man'  would  have  not  personal  but  simple 
supposition.  For  although  this  (concrete)  man  did  not  become 
God,  since  the  suppositum  of  this,  the  person  of  the  Son  of 
God,  was  God  from  eternity,  yet  man,  speaking  universally, 
was  not  always  God. 

This  text  has  the  further  importance  that  it  may  suggest  the  rea- 
son why  the  Scholastics  spoke  of  'personal'  supposition,  this  being 
the  function  exercised  by  a  term  in  standing  for  individuals  or  an 
individual  (suppositum).  For  this  recalls  to  the  mind  of  a  Scholastic 
the  famous  theological  problem  of  the  person  of  Christ,  as  in  27.20. 

The  essentials  of  the  scholastic  doctrine  of  simple  supposition 
may  be  summed  up  thus:  in  the  proposition  lA  is  B',  the  subject  'A' 
has  of  itself  personal  supposition,  i.e.  it  stands  for  the  individuals, 
but  the  predicate  lB'  has  simple  supposition,  i.e.  it  stands  either  for 
a  property  or  a  class.  But  one  can  also  frame  propositions  in  which 
something  is  predicated  of  such  a  property  or  class,  and  then  the 
subject  must  have  simple  supposition.  It  can  be  seen  that  this 
doctrine  deals  with  no  less  a  subject  than  the  distinction  between 
two  logical  types,  the  first  and  second  (cf.  48.21). 

These  simple  but  historically  important  facts  are  complicated  by 
the  scholastic  development  of  two  other  problems  along  with  this 
doctrine.    They    are    (1)    the   problem    of   analysing   propositions, 



whether  they  should  be  understood  in  a  purely  extensions!  fashion, 
or  with  extensional  subject  and  intensional  predicate.  Thomas  and 
Peter,  in  the  texts  cited,  adopt  the  second  position.  We  shall  treat 
this  problem  a  little  more  explicitly  in  a  chapter  on  the  analysis 
of  propositions  (29.02-04).  Then  (2)  there  is  the  problem  of  the 
semantic  correlate  of  a  term  having  simple  supposition.  This  is  a 
very  difficult  philosophical  problem,  and  the  Scholastics  were  of 
varying  opinions  about  its  solution.  In  27.18  Peter  seems  to  think 
that  a  term  with  simple  supposition  stands  for  the  essence  (nature) 
of  the  object.  On  the  other  hand  Ockham  and  his  school  hold  that 
the  semantic  correlate  of  such  a  term  is  simple,  'the  intention  of  the 

27.21  A  term  cannot  have  simple  or  material  supposition 
in  every  proposition  but  only  when  ...  it  is  linked  with  another 
extreme  which  concerns  an  intention  of  the  soul  or  an  utterance 
or  something  written.  E.g.  in  the  proposition  'a  man  is  running' 
the  'man'  cannot  have  simple  or  material  supposition,  since 
'running'  does  not  concern  either  an  intention  of  the  soul,  nor 
an  utterance  nor  something  written.  But  in  the  proposition 
'man  is  a  species'  it  can  have  simple  supposition  because 
'species'  signifies  an  intention  of  the  soul. 

In  this  and  similar  texts  (27.22)  it  is  of  logical  interest  that 
Ockham  and  his  followers  were  apparently  trying  to  give  an  exten- 
sional interpretation  even  to  terms  having  simple  supposition; 
their  correlates  would  be  (concrete)  intentions. 

After  Buridan  there  were  in  the  Middle  Ages,  as  at  the  beginning 
of  the  20th  century,  some  logicians  who  equated  simple  and  material 
supposition.  Paul  of  Venice  gives  that  information : 

27.23  Simple  supposition  is  distinct  from  material  and 
personal ;  some  say  otherwise,  and  make  no  distinction  between 
simple  and  material  supposition.  But  (wide)  it  is  evident  that 
the  subject  does  not  suppose  materially  when  it  is  said:  'the 
divine  essence  is  inwardly  communicable'. 


The  most  usual  supposition  of  a  term  is  personal.  As  Ockham 


27.24  It  is  also  to  be  noticed  that  in  whatever  proposition 
it  be  put,  a  term  can  always  have  personal  supposition,  unless 
it  be  restricted  to  some  other  by  the  will  of  those  who  use  it. 



We  give  the  definition  and  divisions  of  this  kind  of  supposition 
according  to  Peter  of  Spain,  whose  text  contains  the  essentials  of 
the  doctrine  that  remained  standard  till  the  end  of  the  scholastic 

27.25  Personal  supposition  is  the  taking  of  a  common  term 
for  its  inferiors,  as  when  it  is  said  'a  man  runs',  the  term  'man' 
supposes  for  its  inferiors,  viz.  for  Socrates  and  for  Plato  and 
so  on. 

27.26  Of  personal  suppositions  one  kind  is  determinate, 
another  confused.  Determinate  supposition  is  the  taking  of  a 
common  term  put  indefinitely  or  with  the  sign  of  particularity, 
as  'a  man  runs'  or  'some  man  runs',  and  both  are  called 
'determinate'  because  although  in  both  the  term  'man' 
supposes  for  every  man,  whether  running  or  not,  yet  they  are 
true  only  for  one  man  running.  For  it  is  one  thing  to  suppose 
(for  things),  and  another  to  render  the  proposition  true  for 
one  of  them*.  But  as  has  been  said,  the  term  'man'  supposes 
for  all  whether  running  or  not,  yet  renders  the  propositions 
true  only  for  one  who  is  running.  But  it  is  clear  that  the 
supposition  is  determinate  in  both  (propositions),  because 
when  it  is  said:  'An  animal  is  Socrates,  an  animal  is  Plato, 
and  so  on,  therefore  every  animal  is  every  man',  this  is  the 
fallacy  of  the  form  of  speech  (proceeding)  from  a  number  of 
determinates  to  one  (cf.  11.19  and  27.18).  And  so  a  common 
term  put  indefinitely  has  determinate  supposition,  and 
similarly  if  it  has  the  sign  of  particularity. 

27.27  But  confused  supposition  is  the  taking  of  a  common 
term  for  a  number  of  things  by  means  of  the  sign  of  univer- 
sality, as  when  it  is  said:  'every  man  is  an  animal',  the  term 
'man'  is  taken  for  a  number  by  means  of  the  sign  of  uni- 
versality, being  taken  for  each  of  its  individuals. 

Subsequent  division  of  confused  supposition  into  that  which  is 
confused  by  the  requirements  of  the  sign  (necessitate  signi)  and  that 
which  is  confused  by  the  requirements  of  the  thing  (rei)  (27.28)  is 
shortly  after  rejected  by  Peter.  He  gives  a  further  division  of  perso- 
nal supposition: 

27.29  Of  personal  supposition  one  kind  is  restricted, 
another  extended  (ampliata). 

*   Reading  praedictis  for  praedicatis. 




If  we  ask  how  the  expression  'supposition'  is  to  be  rendered  in 
modern  terms,  we  have  to  admit  that  it  cannot  he.  'Supposition' 

covers  numerous  semiotic  functions  for  which  we  now  have  no 
common  name.  Some  kinds  of  supposition  quite  clearly  belong  to 
semantics,  as  in  the  case  of  both  material  suppositions,  and  personal ; 
others  again,  such  as  simple  supposition  and  those  into  which 
personal  supposition  is  subdivided,  are  as  Moody  has  acutely  remark- 
ed (27.30),  not  semantical  but  purely  syntactical  functions. 

The  most  notable  difference  between  the  doctrine  of  supposition 
and  the  corresponding  modern  theories  lies  in  the  fact  that  while 
contemporary  logic  as  far  as  possible  has  one  sign  for  one  function, 
e.g.  a  sign  for  a  word,  another  for  the  word's  name,  one  for  the  word 
in  personal,  another  for  it  in  simple  supposition,  the  Scholastics 
took  equiform  signs  and  determine  their  functions  by  establishing 
their  supposition.  And  this  brings  us  back  to  the  fundamental 
difference  already  remarked  on  between  the  two  forms  of  formal 
logic;  scholastic  logic  dealt  with  ordinary  language,  contemporary 
logic  develops  an  artificial  one. 


Among  the  other  properties  of  terms  three  that  seem  to  be  of  parti- 
cular interest  for  formal  logic  will  be  illustrated  with  some  texts, 
viz.  ampliation,  appellation  and  analogy. 

Peter  of  Spain  writes: 

28.01  Restriction  is  the  narrowing  of  a  common  term  from 
a  wider  (maiore)  supposition  to  a  narrower,  as  when  it  is  said 
'a  white  man  runs'  the  adjective  'white'  restricts  'man'  to 
supposing  for  white  ones.  Ampliation  is  the  extension  (exiensio) 
of  a  common  term  from  a  narrower  supposition  to  a  wider,  as 
when  it  is  said  'a  man  can  be  Antichrist'  the  term  'man' 
supposes  not  only  for  those  who  are  now,  but  also  for  those  who 
will  be.  Hence  it  is  extended  to  future  ones.  I  say  'of  a  common 
term'  because  a  discrete  term  is  neither  restricted  nor  extended. 

One  kind  of  ampliation  is  effected  by  a  verb,  as  by  the 
verb  'can',  e.g.  'a  man  can  be  Antichrist';  another  is  effected 
by  a  name,  e.g.  'it  is  possible  that  a  man  can  be  Antichrist' ; 
another  by  a  participle,  e.g.  'a  man  is  able  (potens)  to  be 
Antichrist' ;  another  by  an  adverb,  e.g.  'a  man  is  necessarily  an 



animal'.  For  (in  the  last)  'man'  is  extended  not  only  for  the 
present  but  also  for  the  future.  And  so  there  follows  another 
division  of  ampliation :  one  kind  of  ampliation  being  in  respect 
of  supposita,  e.g.  'a  man  can  be  Antichrist',  another  with 
respect  to  time,  e.g.  'a  man  is  necessarily  an  animal',  as  has 
been  said. 

Essentially  the  same  doctrine  but  more  thoroughly  developed  is 
found  at  the  end  of  the  14th  century  in  Albert  of  Saxony: 

28.02  Ampliation  is  the  taking  of  a  term  for  one  or  more 
things  beyond  what  is  actually  the  case:  for  that  or  those 
things  for  which  the  proposition  indicates  (denotat)  that  it  is 
used.  Certain  rules  are  established  in  this  respect: 

28.03  The  first  is  this:  every  term  having  supposition  in 
respect  of  a  verb  in  a  past  tense  is  extended  to  stand  for  what 
was,  e.g.  when  it  is  said:  'the  white  was  black',  'the  white'  is 
taken  in  this  proposition  not  only  for  what  is  white  but  for 
what  was  white. 

28.04  Second  rule :  a  term  having  supposition  in  respect  of 
a  verb  in  a  future  tense  is  extended  to  stand  for  what  is  or 
will  be.  .  .  . 

28.05  Third  rule :  every  term  having  supposition  with 
respect  to  the  verb  'can'  is  so  extended  as  to  stand  for  what  is 
or  can  be.  E.g.  'the  white  can  be  black'  means  that  what  is 
white  or  can  be  white,  can  be  black.  .  .  . 

28.06  Fourth  rule :  A  term  having  supposition  in  respect  of 
the  verb  'is  contingent'  is  extended  to  stand  for  what  is  or 
can  contingently  be  (contingit  esse).  And  that  is  Aristotle's 
opinion  in  the  first  book  of  the  Prior  {Analytics).  .  .  . 

28.07  Fifth  rule:  A  term  subjected  in  a  proposition  in 
respect  of  a  past  participle,  even  though  the  copula  of  this 
proposition  is  a  verb  in  the  present,  is  extended  to  stand  for 
what  was.  .  .  .  E.g.  in  the  proposition  'a  certain  man  is  dead' 
the  subject  stands  for  what  is  or  has  been. 

28.08  Sixth  rule :  In  a  proposition  in  which  the  copula  is  in 
the  present,  but  the  predicate  in  the  future,  the  subject  is 
extended  to  stand  for  what  is  or  will  be.  E.g.  'a  man  is  one 
who  will  generate';  for  this  proposition  indicates  that  one 
who  is  or  will  be  a  man  is  one  who  will  generate. 

28.09  Seventh  rule:  If  the  proposition  has  a  copula  in  the 
present  and  a  predicate  that  includes  the  verb  'can',  as  is  the 
case  with  verbal  names  ending  in  '-ble'  ('-ibile'),  then  the  sub- 



ject  is  extended  to  stand  for  what  is  or  can  be,  e.g.  when  it  is 
said:  'the  man  is  generable'.  For  this  is  equivalent  (valet)  to: 
'the  man  can  be  generated'  in  which  'man'  is  extended, 
according  to  the  third  rule,  to  stand  for  what  is  or  can  be.  .  .  . 

28.10  Eighth  rule:  all  verbs  which,  although  not  in  the 
present,  have  it  in  their  nature  to  be  able  to  extend  to  a  future, 
past  or  possible  thing  as  to  a  present  one,  extend  the  terms  to 
every  time,  present,  past  and  future.  Such  e.g.  are  these: 
'I  understand',  'I  know',  'I  am  aware', 'I  mean  (significo)'  etc. 

20.11  Ninth  rule:  the  subject  of  every  proposition  de 
necessario  in  the  divided  sense  (cf.  §  29,  D.)  is  extended  to 
stand  for  what  is  or  can  be.  E.g.  'every  B  is  necessarily  A'; 
for  this  is  equivalent  to  (valet  dicere)  'Whatever  is  or  can  be 
B,  is  necessarily  A'.  .  .  . 

28.12  Tenth  rule:  if  no  ampliating  term  is  present  in  a 
proposition,  its  subject  is  not  extended  but  this  proposition 
indicates  that  (the  subject  stands)  only  for  what  is. 

This  text  is  a  fine  example  of  scholastic  analysis  of  language.  It 
introduces  a  notable  enlargement  of  the  doctrine  of  supposition, 
dividing  the  objects  for  which  a  term  may  stand  into  three  tem- 
poral classes  to  which  is  added  the  class  of  possible  objects.  It 
can  readily  be  seen  that  this  doctrine  makes  an  essential  contri- 
bution to  the  problem  of  the  so-called  void  class,  since  the  expres- 
sion 'void  class'  receives  as  many  different  denotata  as  there  are 
kinds  of  ampliation.  This  can  be  compared  with  the  modern  me- 
thods of  treating  the  problem  (cf.  §  46,  A  and  B). 

Albert's  seventh,  eighth  and  ninth  rules  also  contain  an  analysis 
of  modal  propositions,  but  this  subject  will  be  considered  in  greater 
detail  below  (§  33). 


Closely  connected  with  ampliation  is  the  so-called  appellation, 
also  relevant  to  the  problem  of  the  void  class.  The  theory  of  it  was 
already  well  developed  in  the  13th  century,  was  further  enlarged  in 
the  14th  when  there  were  various  theories  different  from  that  of  the 
13th.*  We  cite  two  13th  century  texts,  one  from  Peter  of  Spain 
and  one  from  Shyreswood: 

28.13  Appellation  is  the  taking  of  a  term  for  an  existent 
thing.  I  say  'for  an  existent  thing'  since  a  term  signifying  a 
non-existent  has  no  appellation,  e.g.  'Caesar'  or  'Anti-christ' 

*  For  information  on  these  points  and  much  other  instruction  on  the  doc- 
trines of  supposition  and  appellation  I  am  obliged  to  Prof.  E.  Moody. 



etc.  Appellation  differs  from  supposition  and  signification  in 
that  appellation  only  concerns  existents,  but  supposition  and 
signification  concern  both  existents  and  non-existents,  e.g. 
'Antichrist'  signifies  Antichrist  and  supposes  for  Antichrist 
but  does  not  name  (appellai)  him,  whereas  'man'  signifies  man 
and  naturally  supposes  for  existent  as  well  as  non-existent 
men  but  only  names  existent  ones. 

Of  appellations  one  kind  belongs  to  common  terms  such  as 
'man',  another  to  singular  terms  such  as  'Socrates'.  A  singular 
term  signifies,  supposes  and  names  the  same  thing,  because  it 
signifies  an  existent,  e.g.  'Peter'. 

Further,  of  appellations  belonging  to  common  terms  one 
kind  belongs  to  a  common  term  (standing)  for  the  common 
thing  itself,  as  when  a  term  has  simple  supposition,  e.g.  when 
it  is  said  :  'man  is  a  species'  or  'animal  is  a  genus' ;  and  then  the 
common  term  signifies,  supposes  and  names  the  same  thing, 
as  'man'  signifies  man  in  general  and  supposes  for  man  in 
general  and  names  man  in  general.  Another  kind  belongs  to  a 
common  term  (standing)  for  its  inferiors,  as  when  a  common 
term  has  personal  supposition,  e.g.  when  it  is  said :  'a  man  runs'. 
Then  'man'  does  not  signify,  suppose  and  name  the  same 
thing;  because  it  signifies  man  in  general  and  supposes  for 
particular  men  and  names  particular  existent  men. 

28.14  Supposition  belongs  to  (inest)  a  term  in  so  far  as  it 
is  under  another.  But  appellation  belongs  to  a  term  in  so  far 
as  it  is  predicable  of  its  (subordinate)  things  by  means  of  the 
verb  'is'.  .  .  .  Some  say  therefore  that  the  term  put  as  subject 
supposes,  and  that  put  as  predicate  names.  ...  It  should  also 
be  understood  that  the  subject-term  names  its  thing,  but  not 
qua  subject.  The  predicate-term  on  the  other  hand  names  it 
qua  predicate. 

The  following  from  Buridan  may  serve  as  an  example  of  14th 
century  theories:* 

28.15  First  it  is  to  be  understood  that  a  term  which  can 
naturally  suppose  for  something  names  all  that  it  signifies 
or  consignifies  unless  it  be  limited  to  what  it  stands  for.  .  .  . 
E.g.  'white'  standing  for  men  names  whiteness,  and  'great' 
greatness,  and  'father'  the  past  (act  of)  generation  and  someone 

*  These  texts  were  communicated  by  Prof.  E.  Moody  who  also  pointed  out 
their  great  importance.  He  is  to  be  thanked  also  for  the  main  lines  of  the  commen- 



else  whom  the  father  has  generated,  and  'the  distant'  names 
that  from  which  it  is  distant  and  the  space  (dimensionem) 
between  them  by  which  it  is  made  distant.  .  .  . 

28.16  A  term  names  what  it  names  as  being  somehow 
determinant  (per  modum  adiacenlis  aliquo  modo)  or  not 
determinant  of  that  for  which  it  stands  or  naturally  can 
stand.  .  .  . 

Thirdly  it  is  to  be  held  that  according  to  the  different 
positive  kinds  of  determination  of  the  things  named  -  the  things 
for  which  the  term  stands  -  there  are  different  kinds  of 
predication,  such  as  how,  how  many,  when,  where,  how  one  is 
related  to  another,  etc.  It  is  from  these  different  kinds  of 
predication  that  the  different  predicaments  are  taken  .  .  . 
(cf.  11.15). 

28.17  Appellative  terms  name  differently  in  respect  of  an 
assertoric  verb  in  the  present  and  in  respect  of  a  verb  in  the 
past  and  in  the  future,  and  in  respect  of  the  verb  'can'  or  of 
'possible';  since  in  respect  of  a  verb  in  the  present  the  appel- 
lative term  -  provided  there  is  not  ampliative  term  -  whether 
it  be  put  as  subject  or  predicate,  names  its  thing  as  something 
connected  with  it  in  the  present,  for  which  the  term  can 
naturally  stand,  and  as  connected  with  it  in  this  or  that  man- 
ner, according  to  which  it  names. 

This  is  a  different  doctrine  from  that  of  the  13th  century,  and 
seems  to  be  of  the  highest  importance.  For  according  to  it  a  term 
does  not  name  what  it  stands  for  but  something  related  to  it  by, 
it  would  seem,  any  relation.  Buridan  says  this  expressly  for  the  term 
'distant'.  If  A  is  distant  from  B  'distant'  does  not  name  A,  but 
precisely  B.  That  indicates  a  clear  notion  of  relation-logic.  Where 
we  should  write  'relation',  Buridan  has  adiacentia.  Especially  im- 
portant is  28.16  where  Buridan  goes  so  far  as  to  say  that  absolute 
terms  are  definable  by  relations,  an  idea  corresponding  to  the  relative 
descriptions  of  47.20.  Some  interesting  results  would  follow  from  the 
detailed  working  out  of  the  basic  notions  of  this  text,  e.g.  a  theory 
of  plural  quantification,  but  we  have  no  knowledge  of  this  being 
done  in  the  Middle  Ages. 


In  the  present  state  of  research  it  is  unfortunately  impossible 
to  present  the  scholastic  theory  of  meaning  with  any  hope  of 
doing  justice  even  only  to  its  essentials.  However,  we  shall  treat  of 
a  further  important  point  in  this  field,  the  theory  of  analogy.  This 



is  of  direct  relevance  to  formal  logic,  and  fairly  well  explored.  A 
single  text  from  Thomas  Aquinas  will  suffice: 

28.18  Nothing  can  be  predicated  univocally  of  God  and 
creatures;  for  in  all  univocal  predication  the  sense  (ratio)  of 
the  name  is  common  to  both  things  of  which  the  name  is 
univocally  predicated  .  .  .  and  yet  one  cannot  say  that  what  is 
predicated  of  God  and  creatures  is  predicated  purely  equivo- 
cally. ...  So  one  must  say  that  the  name  of  wisdom  is  predi- 
cated of  God's  wisdom  and  ours  neither  purely  univocally 
nor  purely  equivocally,  but  according  to  analogy,  by  which  is 
just  meant:  according  to  a  proportion.  But  conformity 
(convenientia)  according  to  a  proportion  can  be  twofold,  and 
so  a  twofold  community  of  analogy  is  to  be  taken  account  of. 
For  there  is  a  conformity  between  the  things  themselves  which 
are  proportioned  to  one  another  in  having  a  determinate 
distance  of  some  other  relationship  (habitudinem)  to  one  ano- 
ther, e.g.  (the  number)  2  to  unity,  2  being  the  double.  But  we 
also  sometimes  take  account  of  conformity  between  two  things 
which  are  not  mutually  proportioned,  but  rather  there  is  a 
likeness  between  two  proportions;  e.g.  6  is  conformed  to  4 
because  as  6  is  twice  3  so  4  is  twice  2.  The  first  conformity  then 
is  one  of  proportion,  but  the  second  of  proportionality.  So  it  is 
then  that  according  to  the  first  kind  of  conformity  we  find 
something  predicated  analogically  of  two  things  of  which 
one  has  a  relationship  to  the  other,  as  being  is  predicated  of 
substance  and  of  accident  owing  to  the  relationship  which 
substance  and  accident  have  (to  one  another),  and  health  is 
predicated  of  urine  and  animals,  since  urine  has  some  relation- 
ship *  to  the  health  of  animals.  But  sometimes  predication  is 
made  according  to  the  second  kind  of  conformity,  e.g.  the 
name  of  sight  is  predicated  of  corporeal  sight  and  of  intellect, 
because  as  sight  is  in  the  eye,  so  intellect  is  in  the  mind. 

This  is  about  the  clearest  text  of  the  many  in  which  Thomas 
Aquinas  speaks  of  analogy  (28.19).  It  has  been  only  too  often 
misunderstood,  but  deserves  fairly  thorough  discussion  from  the 
historian  of  logic  because  of  its  historical  as  well  as  systematic 
significance.   We  therefore  draw  attention  to  the  following  points: 

This  text  deals  explicitly  with  a  question  of  semantics  -  Thomas 
speaks  of  names  -  and  it  is  noteworthy  that  he  himself,  like  his  best 
commentator  Cajetan  (28.20),  almost  always  considers  analogy  as 

*   Reading  habitudinem  for  similitudinem. 



'of  names'.  Of  course  he  does  not  mean  mere  utterances,  but  mean- 
ingful words,  in  accordance  with  the  scholastic  usage  illustrated 


Now  our  text  speaks  of  three  classes  of  names:  univocal,  equivo- 
cal and  analogous  names.  The  last  are  intermediate  between  the 
two  first.  The  class  of  analogous  names  falls  into  two  sub-classes: 
those  analogous  according  to  a  proportion,  and  those  according  to 
proportionality.  Both  these  divisions  originate  with  Aristotle 
(10.29  and  10.31),  but  the  hasty  indications  of  the  Nichomachean 
Ethics  are  here  developed  into  a  systematic  logical  doctrine. 

While  the  thomist  doctrine  of  the  first  class  of  analogous  names 
is  here  only  of  interest  as  showing  an  attempt  to  formalize  the  rules 
of  their  use,  the  theory  of  the  second  class,  i.e.  of  names  analogous 
according  to  proportionality,  is  nothing  less  than  a  first  formulation 
of  the  notion  of  isomorphy  (cf.  47.41).  That  this  is  so  can  be  seen  as 
follows : 

Let  us  note  first  that  according  to  the  text  an  analogous  name  of 
the  second  kind  always  refers  to  a  relation  or  relata  defined  by  one. 
Certainly  something  absolute  is  also  implied  by  each  of  the  subjects 
in  such  an  analogy,  but  this  is  precisely  different  in  each,  and  in 
that  respect  the  name  is  equivocal.  The  community  of  reference 
consists  only  in  regard  to  certain  relations. 

But  it  is  not  a  matter  of  just  one  relation,  rather  of  two  similar 
ones.  This  is  explicit  in  the  text,  only  the  example  (6:3  =  4:2)  is 
misleading  since  we  have  there  an  identity  of  two  relations.  That 
Thomas  is  not  thinking  of  such  is  shown  by  the  illustrations,  first  in 
the  domain  of  creatures  (sight: eye  —  intellect: mind),  then  in  God 
(divine  being: God  -  creaturely  being: creature).  The  ruling  idea  is 
then  of  a  relation  of  similarity  between  two  relations. 

This  relation  between  relations  is  such  as  to  allow  inference  from 
what  we  know  about  one  to  something  about  the  other,  though  at 
the  same  time  we  have  the  assertion :  'we  cannot  know  what  God  is' 
(28.21).  The  apparent  contradiction  disappears  when  it  is  realized 
that  we  are  dealing  with  isomorphy.  For  this  does  in  fact  allow  one 
to  transfer  something  from  one  relation  to  another,  without  afford- 
ing any  experience  of  the  relata. 

The  use  of  a  mathematical  example  is  noteworthy,  taken  more- 
over from  the  only  algebraic  function  then  known.  This  is  not  only  to 
be  explained  by  the  mathematical  origin  of  the  doctrine  of  analogy 
in  Aristotle,  but  also  perhaps  by  a  brilliant  intuition  on  the  part  of 
Aquinas  who  dimly  guessed  himself  to  be  establishing  a  thesis  about 
structure.  In  any  case  the  text  is  of  the  utmost  historical  importance 
as  being  the  first  indication  of  a  study  of  structure,  which  was  to 
become  a  main  characteristic  of  modern  science. 




We  give  first  a  text  of  Albert  of  Saxony  which  summarizes  the 
commonly  received  scholastic  doctrine  of  the  kinds  of  atomic 
(categorical)  propositions : 

29.01  Of  proposition  some  are  categorical,  others  hypo- 
thetical. But  some  of  the  categorical  are  said  to  be  hypothetical 
in  signification,  such  as  the  exclusive,  exceptive  and  redupli- 
cative propositions,  and  others  besides. 

Then  of  the  categorical  propositions  that  are  not  equi- 
valent to  the  hypothetical  in  signification  -  such  as  'man  is  an 
animal'  and  such-like  -  some  are  said  to  be  assertoric  (de 
inesse)  or  of  simple  inherence;  others  are  said  to  be  modal  or 
of  modified  inherence.  .  .  . 

Again  of  categorical  propositions  of  simple  inherence  some 
have  ampliative  subjects,  as  'a  man  is  dead',  'Antichrist  will 
exist',  the  others  do  not  have  ampliative  subjects,  as  'man  is  an 
animal',  'stone  is  a  substance'  etc. 

Again,  of  categorical  propositions  of  simple  inherence 
with  ampliative  subjects,  some  concern  the  present,  others  the 
past,  others  the  future.  .  .  . 

Again,  of  categorical  propositions  about  the  present  some 
are  de  secundo  adiacente,  others  de  tertio  adiacente.  An  example 
of  the  first :  'man  exists' ;  of  the  second :  'man  is  an  animal'. 

Again,  of  categorical  propositions  some  have  a  non-com- 
pound extreme  (term)  (de  extremo  incomplexo),  as  'man  is  an 
animal';  others  have  a  compound  extreme,  as  'man  or  ass  is 
man  or  ass'. 


Here  we  assemble  a  few  aspects  of  the  scholastic  analysis  of 
propositions.  To  begin  with,  this  text  of  Thomas  Aquinas,  followed 
by  one  from  Ockham,  about  the  general  structure: 

29.02  In  every  true  affirmative  proposition  the  subject 
and  predicate  must  signify  somehow  the  same  thing  in  reality 
but  in  different  senses  (diversum  secundum  rationem).  And  this 
is  clear  both  in  propositions  with  accidental  predicate  and  in 
those  with  substantial.  For  it  is  evident  that  'man'  and 
'white'  are  identical  in  suppositum  and  differing  in  sense,  for 
the  sense  of  'man'  is  other  than  the  sense  of  'white'.  And 



likewise  when  I  say:  'man  is  an  animal',  for  that  same  thing 
which  is  man  is  truly  an  animal.  For  in  one  and  the  same 
suppositum  there  is  both  the  sensible  nature,  after  which  it  is 
called  'animal',  and  the  rational  nature,  after  which  it  is  called 
'man'.  So  that  in  this  case  too  the  predicate  and  subject  are 
identical  as  to  suppositum,  but  differing  in  sense.  But  this  is 
also  found  in  a  way  in  propositions  in  which  something  (idem) 
is  predicated  of  itself,  inasmuch  as  the  intellect  treats  as 
suppositum  (trahit  ad  partem  suppositi)  what  it  posits  as 
subject,  but  treats  as  form  inhering  in  the  suppositum  what  it 
posits  as  predicate.  Hence  the  adage,  that  predicates  are 
taken  formally  and  subjects  materially  (cf.  27.08).  To  the 
difference  in  sense  there  corresponds  the  plurality  of  predi- 
cate and  subject;  but  the  intellect  signifies  the  real  identity  by 
the  composition  (of  the  two). 

We  have  here  actually  two  analyses  of  propositions.  First  an 
extensional  one,  which  seems  to  have  become  classical  in  later 
Scholasticism.  It  can  be  reproduced  thus:  the  proposition  'S  is  P' 
is  to  be  equated  with  the  product  of  the  following  propositions: 
(1)  there  is  at  least  one  x  such  that  both  lS'  and  'P'  stand  (suppose) 
for  x,  (2)  there  is  a  property  /  such  that  'S'  signifies  /,  (3)  there  is  a 
property  g  such  that  'P'  signifies  g,  (4)  both  /  and  g  belong  to  x. 

In  the  second  analysis  the  subject  is  conceived  as  extensional,  the 
predicate  as  intensional.  The  proposition  'A  =  A'  can  be  interpreted : 
(1)  there  is  an  x  such  that  'A'  stands  for  x,  (2)  there  is  a  property  / 
such  that  'A'  signifies  /,  (3)  /  belongs  to  x.  This  analysis  is  applied  in 
the  text  to  a  special  kind  of  proposition,  asserting  an  identity,  but 
can  evidently  be  applied  generally. 

Ockham  gives  another  analysis : 

29.03  It  is  to  be  said  that  it  is  not  required  for  the  truth  of 
a  singular  proposition,  which  is  not  equivalent  to  many 
propositions,  that  subject  and  predicate  should  be  really 
identical,  nor  that  the  predicated  reality  should  be  in  the 
subject,  nor  that  it  should  really  inhere  (insit)  in  the  subject, 
nor  that  it  should  be  really,  extra-mentally,  united  to  the 
subject.  E.g.  it  is  not  required  for  the  truth  of  this:  'that  one 
is  an  angel',  that  the  common  term  'angel'  should  be  in  reality 
the  same  as  what  is  posited  as  subject,  nor  that  it  should 
really  inhere  in  it,  nor  anything  of  that  kind,  -  but  it  is 
sufficient  and  necessary  that  subject  and  predicate  should 
suppose  for  the  same  thing.  And  so  in  this :  'this  is  an  angel',  if 
subject  and  predicate  suppose  for  the  same  thing,  the  pro- 



position  is  true.  And  so  it  is  not  indicated  (denotatur)  that  this 
has  angelicity,  or  that  angelicity  is  in  it,  or  anything  of  this 
kind,  but  it  is  indicated  that  this  is  truly  an  angel,  not  that  it 
is  that  predicate,  but  that  it  is  that  for  which  the  predicate 

An  important  text,  but  not  readily  intelligible  to  a  modern  reader. 
A  possible,  though  not  the  only  possible  interpretation  is  this:  it  is 
sufficient  and  necessary  for  the  truth  of  a  proposition  of  this  kind 
that  the  extension  of  subject  and  predicate  should  coincide.  If  that 
is  so,  he  means  to  say  that  the  predicate  is  not  to  be  taken  intension- 
ally,  but  extensionally  like  the  subject,  as  in  Thomas's  first  ana- 
lysis in  29.02.  Then  Ockham  gives  a  radically  extensional  inter- 
pretation of  propositions. 

The  next  text  shows  that  what  was  said  in  29.03  holds  for  other 
kinds  of  proposition  as  well: 

29.04  For  the  truth  of  such  (i.e.  indefinite  or  particular 
propositions)  it  suffices  that  the  subject  and  predicate  stand 
for  the  same  thing,  if  the  proposition  is  affirmative. 


In  the  middle  of  the  13th  century  there  arose  a  generally  accepted 
doctrine  about  the  structure  of  modal  propositions.  It  is  to  be  found 
in  Albert  the  Great  (29.05),  Shyreswood  (29.06),  Peter  of  Spain  (29.07), 
and  in  the  Summa  Toiius  Logicae  (29.08).  On  account  of  its  charac- 
teristic formalism  we  quote  a  youthful  opusculum  of  Thomas 

29.09  Since  the  modal  proposition  gets  its  name  from 
'modus',  to  know  what  a  modal  proposition  is  we  must  know 
what  a  modus  is.  Now  a  modus  is  a  determination  of  something 
effected  by  a  nominal  adjective  determining  a  substantive, 
e.g.  'white  man',  or  by  an  adverb  determining  a  verb.  But  it 
is  to  be  known  that  modes  are  threefold,  some  determining 
the  subject  of  a  proposition,  as  'a  white  man  runs',  some  deter- 
mining the  predicate,  as  'Socrates  is  a  white  man'  or  'Socrates 
runs  quickly',  some  determining  the  composition  of  the 
predicate  with  the  subject,  as  'that  Socrates  is  running  is 
impossible',  and  it  is  from  this  alone  that  a  proposition  is  said 
to  be  modal.  Other  propositions,  which  are  not  modal,  are  said 
to  be  assertoric  (de  inesse). 

The  modes  which  determine  the  composition  are  six:  'true', 
'false',   'necessary',   'possible',   'impossible'  and  'contingent'. 



But  'true'  and  'false'  add  nothing  to  the  signification  of  asser- 
toric  propositions;  for  there  is  the  same  significance  in 
'Socrates  runs'  and  it  is  true  that  Socrates  runs'  (on  the  one 
hand),  and  in  'Socrates  is  not  running'  and  'it  is  false  that 
Socrates  is  running'  (on  the  other).  This  does  not  happen  with 
the  other  four  modes,  because  there  is  not  the  same  significance 
in  'Socrates  runs'  and  'that  Socrates  runs  is  impossible  (or 
necessary)'.  So  we  leave  'true'  and  'false'  out  of  consideration 
and  attend  to  the  other  four. 

Now  because  the  predicate  determines  the  subject  and  not 
conversely,  for  a  proposition  to  be  modal  the  four  modes 
aforesaid  must  be  predicated  and  the  verb  indicating  com- 
position must  be  put  as  subject.  This  is  done  if  an  infinitive  is 
taken  in  place  of  the  indicative  verb  in  the  proposition, 
and  an  accusative  in  place  of  the  nominative.  And  it  (the 
accusative  and  infinitive  clause)  is  called  'dictum',  e.g.  of  the 
proposition  'Socrates  runs'  the  dictum  is  'that  Socrates  runs' 
(Socratem  currere).  When  then  the  dictum  is  posited  as  subject 
and  a  mode  as  predicate,  the  proposition  is  modal,  e.g.  'that 
Socrates  runs  is  possible'.  But  if  it  be  converted  it  will  be 
assertoric,  e.g.  'the  possible  is  that  Socrates  runs'. 

Of  modal  propositions  one  kind  concerns  the  dictum, 
another  concerns  things.  A  modal  (proposition)  concerning 
the  dictum  is  one  in  which  the  whole  dictum  is  subjected  and 
the  mode  predicated,  e.g.  'that  Socrates  runs  is  possible'.  A 
modal  (proposition)  concerning  things  is  one  in  which  the 
mode  interrupts  the  dictum,  e.g.  'for  Socrates  running  is 
possible'  (Socratem  possibile  est  currere).  But  it  is  to  be  known 
that  all  modals  concerning  the  dictum  are  singular,  the  mode 
being  posited  as  inherent  in  this  or  that  as  in  some  singular 
thing.  But  .  .  .  modals  concerning  things  are  judged  to  be 
universal  or  singular  or  indefinite  according  to  the  subject  of 
the  dictum,  as  is  the  case  with  assertoric  propositions.  So 
that  'for  all  men,  running  is  possible'  is  universal,  and  so 
with  the  rest.  It  should  further  be  known  that  modal  pro- 
positions are  said  to  be  affirmative  or  negative  according 
to  the  affirmation  or  negation  of  the  mode,  not  according  to 
the  affirmation  or  negation  of  the  dictum.  So  that  .  .  .  this 
modal  'that  Socrates  runs  is  possible'  is  affirmative,  while 
'that  Socrates  runs  is  not  possible'  is  negative. 

There  are  two  notable  points  in  this  text.  First  there  is  the  very 
thorough  formalism,  the  modal  proposition  being  classified  accord- 



ing  to  the  place  which  the  mode  has  in  it.  Then  there  is  the  explicit 
distinction  of  the  two  structures,  one  of  which  Aristotle  made  the 
basis  of  his  modal  logic  (§  15,  B),  the  other  of  which  Theophrastus 
adopted  (§  17,  B).  The  modals  de  re  correspond  to  the  Aristotelian 
structure,  in  which  the  mode  does  not  determine  the  'composition' 
or,  as  we  should  say,  the  proposition  as  a  whole,  but  'the  predicate'. 
Taking  the  proposition  A  is  possibly  BJ  as  de  re,  we  could  analyze 
it  thus: 

if  x  is  A,  then  x  is  possibly  B. 
But  the  modals  de  dido  have  the  Theophrastan  structure,  according 
to  which  the  fore-going  proposition  will  be  taken  as  de  dido  and  can 
be  interpreted: 

that  A  is  B  is  possible. 


Closely  connected  with  that  doctrine,  classical  in  Scholasticism, 
is  that  of  the  composite  and  divided  senses  of  propositions.  It  was 
developed  out  of  the  Aristotelian  theory  of  the  fallacies  of  division 
and  composition  (11.22f.),  and  partly  corresponds  to  the  foregoing 
analysis  of  modal  propositions  (29.09),  but  extends  to  other  kinds 
as  well.  It  seems  to  have  secured  a  quite  central  place  in  later 
scholastic  logic.  We  cite  first  a  text  of  Peter  of  Spain: 

29.10  There  are  two  kinds  of  composition.  The  first  kind 
arises  from  the  fact  that  some  dictum  can  suppose  for  itself 
or  a  part  of  itself,  e.g.  'that  he  who  is  sitting  walks  is  possible'. 
For  if  the  dictum  'that  he  who  is  sitting  walks'  is  wholly 
subjected  to  the  predicate  'possible',  then  the  proposition  is 
false  and  composite,  for  then  opposed  activities,  sitting  and 
walking,  are  included  in  the  subject,  and  the  sense  is:  'he 
who  is  sitting  is  walking'.  But  if  the  dictum  supposes  for  a  part 
of  the  dictum,  then  the  proposition  is  true  and  divided,  and 
the  sense  is:  'he  who  is  sitting  has  the  power  of  walking'.  To 
be  distinguished  in  the  same  fashion  is :  'that  he  who  is  not 
writing  is  writing  is  impossible'.  For  this  dictum  'that  he 
who  is  not  writing  is  writing'  is  subjected  to  the  predicate 
'impossible',*  but  sometimes  as  a  whole,  sometimes  in  respect 
of  a  part  of  itself.  And  similarly:  'that  a  white  thing  is  black 
is  possible'.  And  it  is  to  be  known  that  expressions  of  this 
kind  are  commonly  said  to  be  de  re  or  de  dicio. 

Reading  impossibile  for  possibile. 



A  twofold  terminology  can  be  seen  here ;  the  distinction  composita- 
divisa  corresponds  to  de  dido  -  de  re.  Peter  also  introduces  the  notion 
of  supposition,  while  Thomas  (29.09)  proceeds  wholly  syntactically. 
Thomas  has  yet  other  expressions  for  the  same  idea: 

29.11  Further  (it  is  objected),  if  everything  is  known  by 
God  as  seen  in  the  present,  it  will  be  necessary  that  what  God 
knows,  is,  as  it  is  necessary  that  Socrates  sits  given  that  he  is 
seen  to  be  sitting.  But  this  is  not  necessary  absolutely,  or  as  is 
said  by  some,  by  necessity  of  the  consequent:  rather  condi- 
tionally, or  by  necessity  of  consequence.  For  this  conditional 
is  necessary:  If  he  is  seen  to  be  sitting,  he  sits.  Whence  also,  if 
the  conditional  is  turned  into  a  categorical,  so  that  it  is  said: 
what  is  seen  to  be  sitting,  necessarily  sits,  evidently  if  this  is 
understood  as  de  dido  and  composite,  it  is  true ;  but  understood 
as  de  re  and  divided,  it  is  false.  And  so  in  these  and  all  similar 
cases  .  .  .  people  are  deceived  in  respect  of  composition  and 

This  gives  us  the  two  following  series  of  expressions,  correspond- 
ing member  to  member  (the  word  propositio  being  understood  with 
each) :  de  dido,  composita,  necessaria  necessitate  consequentiae, 
necessaria  sub  conditione  -  de  re,  divisa,  necessaria  necessitate  conse- 
queniis,  necessaria  absolute. 

Paul  of  Venice  gives  a  peculiar  variant  of  the  doctrine  of  de  dido 
and  de  re: 

29.12  Some  say  that  always  when  the  mode  simply  precedes 
or  follows  the  expression  with  the  infinitive,  then  the  sense  is 
definitely  called  'composite'  in  every  case,  e.g.  'it  is  possible 
that  Socrates  runs',  'that  Socrates  runs  (Socratem  currere)  is 
possible'.  But  when  the  mode  occupies  a  place  in  the  middle  the 
sense  is  called  'divided',  e.g.  'for  Socrates  it  is  possible  to 
run'.  Others  on  the  other  hand  say  that  when  the  mode 
simply  precedes,  the  sense  is  composite,  as  previously,  but 
when  it  occupies  a  middle  place  or  comes  at  the  end,  then  the 
sense  is  divided,  e.g.  'of  A  I  know  that  it  is  true',  'that  A  is 
true  is  known  by  me'.  And  so  with  others  similar. 

But  though  these  ways  of  speaking  enjoy  probability,  yet 
they  are  not  wholly  true.  ...  So  I  say  otherwise,  taking  a 
position  intermediate  between  them:  when  the  mode  simply 
precedes  a  categorical  or  hypothetical  dictum,  it  effects  the 
composite  sense;  and  when  it  occurs  between  the  verb  and 
the  first  extreme,  it  is  taken  in  the  divided  sense;  but  when  it 



follows  at  the  end,  it  can  be  taken  in  the  composite  or  the 
divided  sense. 

This  seemingly  purely  grammatical  text  is  yet  not  without 
interest  as  showing  how  scholastic  logic  at  the  end  of  the  14th 
century  was  wholly  bent  on  grasping  the  laws  of  everyday  language. 
We  find  no  essentially  new  range  of  problems  in  Paul  beyond  those 
of  Thomas  and  Peter. 

There  is  yet  another  interpretation  of  the  composite  and  divided 
senses,  first  found  in  Peter,  in  a  text  which  seems  to  adumbrate  all 
the  associated  problems: 

29.13  (The  fallacy  of)  division  is  a  false  division  of  things 
that  should  be  compounded.  There  are  two  kinds  of  division. 
The  first  arises  from  the  fact  that  a  conjunction  can  conjoin 
either  terms  or  propositions,  e.g.  .  .  .  Tive  is  even  or  odd'. 
Similarly:  'every  animal  is  rational  or  irrational'.  For  if  this 
conjunctive  particle  'or'  divides  one  proposition  from  another, 
it  is  false,  and  its  sense  is:  'every  animal  is  rational  or  every 
animal  is  irrational'.  If  it  disjoins  one  term  from  another, 
then  it  is  true  and  its  sense  is:  'every  animal  is  rational  or 
irrational',  in  which  the  whole  disjunctive  complex  is  predi- 
cated. Similarly:  'every  animal  is  healthy  or  sick',  'every 
number  is  even  or  odd'. 

A  more  exact  formulation  of  the  same  thought  occurs  in  Burleigh : 

29.14  'Every  animal  is  rational  or  irrational.'  The  proof  is 
inductive.  The  disproof  runs:  every  animal  is  rational  or 
irrational,  but  not  every  animal  is  rational,  therefore  every 
animal  is  irrational.  The  conclusion  is  false,  the  minor  is  not 
(false),  therefore  the  major  is.  The  consequence  is  evident 
from  the  locus  of  opposites  (16.18). 

Solution.  The  first  (proposition)  is  multiple,  according  to 
composition  and  division.  In  the  sense  of  composition  it  is 
true,  in  the  sense  of  division  it  is  false.  Induction  does  not 
hold  in  the  sense  of  division,  because  in  the  sense  of  division 
there  is  not  a  categorical  proposition  but  a  hypothetical  of 
universal  quantity;  and  thus  the  answer  to  the  proof  is 

To  the  disproof,  I  say  the  consequence  does  not  hold  in  the 
sense  of  composition,  nor  is  there  room  for  an  argument  from 
the  locus  of  opposites,  for  the  locus  of  opposites  is  when  one 
argues  from  a  disjunctive  and  the  negation  of  one  part,  to 



the  other  part;  but  in  the  sense  of  composition  this  is  not  a 
disjunctive  but  a  categorical  proposition. 

The  form  of  propositions  of  this  kind  in  the  composite  sense  could 

be  expressed  with  variables  thus: 

(1)  for  every  x:  x  is  A  or  a;  is  B. 
In  the  divided  sense  the  same  proposition  could  be  interpreted  : 

(2)  Tor  every  x:  x  is  A,  or  for  every  x:  x  is  B. 
If  this  interpretation  is  correct  we  have  here  an  important  theorem 
about  the  distribution  of  quantifiers.  Yet  Burleigh  does  Dot  seem 
to  have  been  thinking  quite  of  (1),  but  rather  of 

(V)  for  every  x:  x  is  (A  or  B). 


Finally  we  give  a  text  in  which  the  chief  scholastic  theories  about 
the  semantic  correlate  of  propositions  are  listed.  It  is  taken  from 
Paul  of  Venice. 

29.15  About  the  essence  of  the  proposition  .  .  .  there  are 
many  opinions. 

The  first  is  that  the  significate  of  a  true  proposition  is  a 
circumstance  (modus)  of  the  thing  and  not  the  thing  itself.  .  .  . 

29.16  The  second  opinion  is  that  the  significate  of  a  true 
proposition  is  a  composition  of  the  mind  (mentis)  or  of  the 
intellect  which  compounds  or  divides.  .  .  . 

29.17  The  third  opinion,  commonly  received  among  the 
doctors  of  my  (Augustinian)  Order,  in  particular  by  Master 
Gregory  of  Rimini,  is  that  the  significate  of  a  proposition  is 
whatever  in  any  way  exists  as  a  signifiable  complex.  And 
when  it  is  asked  whether  such  a  signifiable  is  something  or 
nothing,  he  answers  that  the  name  'something'  and  its  syno- 
nyms 'thing'  and  'being'  can  be  understood  in  three  ways. 
(1)  First  in  the  widest  sense,  according  to  which  everything 
signifiable,  with  or  without  complexity,  truly  or  falsely,  is 
called  'thing  '  and  'something'.  ...  (2)  In  a  second  way  these 
(names)  are  taken  for  whatever  is  signifiable,  with  or  without 
complexity,  but  truly,  ...  (3)  In  a  third  way  the  aforesaid 
names  are  taken  in  such  wise  that  they  signify  some  existent 
essence  or  entity,  and  in  this  way,  what  does  not  exist  is 
called  'nothing'  ....  So  this  opinion  says  that  the  significate 
of  a  proposition  is  something,  if  one  takes  the  afore-mentioned 
terms  in  the  first  or  second  way.  .  .  . 

29.18  The  fourth  opinion  posits  some  theses.  (1)  The  first 
is  this :  that  no  thing  is  the  adequate  or  total  significate  of  a 



mental  proposition  properly  so  called;  since  every  such 
(proposition)  signifies  a  variety  of  mutually  distinct  things, 
by  reason  of  its  parts  to  which  it  is  equivalent  in  its  signifying. 
And  this  is  evident  to  everyone  who  examines  the  matter. 
Hence  there  is  no  total  or  adequate  significate  of  such  a 

(2)  The  second  thesis:  whatever  is  signified  by  a  mental 
proposition  properly  so  called  according  to  its  total  significa- 
tion is  also  signified  by  any  of  its  parts.  .  .  . 

(3)  Third  thesis:  no  dictum  corresponding  to  a  mental 
proposition  properly  so  called,  e.g.  an  expression  in  the 
infinitive  mood  taken  as  significant,  supposes  for  any  thing. 
For  instance,  if  the  dictum,  i.e.  the  expression  in  the  infinitive 
mood,  'that  man  is  an  animal',  corresponding  to  the  proposi- 
tion 'man  is  an  animal',  is  taken  materially  it  stands  for  some 
thing,  namely  for  the  proposition  to  which  it  corresponds; 
but  if  it  be  taken  significatively,  i.e.  personally,  then  according 
to  the  fourth  opinion  it  stands  for  no  thing.  This  is  evident, 
since  such  an  expression,  so  taken,  signifies  a  number  of 
things,  viz.  all  those  signified  by  the  corresponding  proposition, 
and  so  there  would  be  no  reason  for  it  to  suppose  for  one  of 
its  significates  rather  than  another;  hence  (it  supposes) 
either  for  each  or  for  none.  But  nobody  would  say  for  each, 
since  the  expression  'that  man  is  an  animal'  would  signify 
an  ass  or  suppose  for  an  ass.  Therefore  for  none.  And  what  is 
said  of  that  instance,  holds  for  any  other. 

The  four  opinions  there  enumerated  could  be  summed  up  thus 
in  modern  terminology;  a  proposition  has  for  its  semantical  correlate 
(1)  a  real  fact,  (2)  a  psychical  act  (  3)  an  objective  content  (the  Stoic 
lekton),  (4)  nothing  at  all  beyond  what  its  parts  already  signify. 
In  the  15th  century  there  were  very  complicated  and  sharp  disputes 
about  this  problem.  But  as  it  lies  on  the  border-line  of  pure  logic 
we  shall  omit  consideration  of  them  here  (29.19). 





The  theory  of  consequences  is  one  of  the  most  interesting  scholastic 
doctrines.  Essentially  it  is  a  development  of  Stoic  prepositional 
logic,  though  so  far  as  is  known  it  was  constructed  entirely  anew, 
not  in  connection  with  the  Stoic  logoi  (§  21 )  but  with  certain  passages 
of  the  Hermeneia  and,  above  all,  the  Topics.  All  the  same,  fragments 
of  the  Stoic  propositional  logic  did  influence  the  Scholastics,  mostly 
through  the  mediation  of  Boethius,  though  for  a  few  we  must 
suppose  some  other  sources,  as  e.g.  for  the  'dog-syllogism'  (22.19;, 
which  is  found  in  Thomas  Aquinas  (30.01).  But  that  these  fragments 
were  not  the  starting  point  is  clear  from  the  fact  that,  at  least  to 
begin  with,  they  are  not  cited  in  the  treatise  on  consequences,  but 
in  another  on  hypothetical  syllogisms. 

The  name  ' consequential'  is  Boethius's  translation  of  Aristotle's 
axoXoi>07](7!.<;  which  occurs  frequently  in  the  Hermeneia  (30.02)  but 
not  in  any  exact  technical  sense,  rather  for  following  quite  in  general. 
The  word  has  the  same  sense  in  Abelard,  though  limited  to  logical 
relationships  between  terms  (30.03),*  and  to  some  extent  also  in 
Kilwardby  *  *  (30.04)  and  Peter  of  Spain  (30.05)  E.g.  in  the  latter  we 
read  of  a  consequenlia  esseniiae  (30.06). 

In  Ockham,  on  the  other  hand,  and  his  successors  the  word 
has  a  sharply  defined  technical  meaning,  and  signifies  a  relation  of 
consequence  between  two  propositions. 

The  following  text  from  Kilwardby  may  serve  as  a  good  example 
of  the  earlier  stage: 

30.04  He  (Aristotle)  also  says  that  something  is  a  consequent 
(of  something  else)  in  part,  and  yet  whatever  follows  from  A 
follows  from  all  that  is  contained  under  A,  since  what  follows 
on  the  consequent  follows  on  the  antecedent,  and  so  every 
consequent  follows  on  the  whole  antecedent.  .  .  . 

It  is  to  be  answered  to  this,  that  (Aristotle)  in  this  whole 
treatment  (Prior  Analytics)  takes  'consequent'  for  the  pre- 
dicate and  'antecedent'  for  the  subject.  .  .  . 

A  further  passage  from  Kilwardby  is  extremely  instructive  about 
this  relationship: 

Vide  Translator's  Preface,  B. 

Cited  from  transcriptions  of  two  Oxford  MSS  (30.04)  made  by  the  translator. 



30.07  Consequence  is  twofold,  viz.  essential  or  natural, 
as  when  a  consequent  is  naturally  understood  in  its  antecedent, 
and  accidental  consequence.  Of  the  latter  kind  is  the  conse- 
quence according  to  which  we  say  that  the  necessary  follows 
on  anything.  .  .  . 

That  shows  that  for  Kilwardby  a  'natural'  and  'essential'  conse- 
quence is  only  present  when  it  is  a  matter  of  connection  between 
terms.  Thus  for  him  the  proposition  'every  man  runs,  therefore 
there  is  a  man  who  runs'  would  be  natural,  since  'each  man'  is 
'naturally'  included  in  'every  man'.  In  other  words  it  is  for  him 
always  based  on  term-logical  relationships.  He  does  also  recognize 
purely  propositional  consequences  such  as  the  one  he  states:  'the 
necessary  follows  on  anything',  but  these  he  considers  only  'acciden- 
tal' and  of  an  inferior  kind. 

This  opinion  of  Kilwardby's  is  of  interest  as  showing  that  the 
Scholastics  did  not  take  the  abstract  propositional  logic  of  the  Stoics 
as  their  starting  point,  but  the  term-logic  of  Aristotle.  Yet  before 
very  long  they  built  on  that  basis  a  technically  excellent  pure  logic 
of  propositions  that  to  the  best  of  our  knowledge  was  superior  to 
that  of  the  Megarians  and  Stoics. 

Since  the  paper  of  Lukasiewicz  (30.08)  more  works  have  been 
devoted  to  this  propositional  logic  than  to  any  other  scholastic 
logical  doctrine,  and  it  is  better  known  than  most  others  (30.09).  Yet 
we  are  still  far  from  having  a  complete  knowledge  of  it.  We  cannot 
here  enumerate  all  the  scholastic  consequences  that  have  been 
investigated  in  the  20th  century,  but  must  limit  ourselves  to  texts 
defining  the  notion  of  consequentia,  and  then  (§  31)  give  a  few  examp- 


Pseudo-Scotus  gives  the  following  definition  of  consequence: 

30.10  A  consequence  is  a  hypothetical  proposition  composed 
of  an  antecedent  and  consequent  by  means  of  a  conditional 
connective  or  one  expressing  a  reason  (rationalis)  which 
signifies  that  if  they,  viz.  the  antecedent  and  consequent, 
are  formed  simultaneously,  it  is  impossible  that  the  ante- 
cedent be  true  and  the  consequent  false. 

Here  a  consequence  is  conceived  as  a  proposition  in  almost  word 
for  word  agreement  with  the  Stoic  definition  (19.15)  with  only  two 
considerable  differences:  (1)  'proposilio'  means,  not  the  lekton,  but 
the  thought,  written  and  spoken  proposition  (cf.  26.03) ;  (2)  the 
consequence  corresponds  to  the  compound  and  inferential  sentences 
of  the  Stoics  (19.15).  Implication  is  defined  in  the  Diodorean  way 



(20.08),  though  the  time-variables  might  be  thought  to  be  missing; 
but  comparison  with  the  definition  of  consequence  ut  nunc  (30.12, 
cf.  30.16)  shows  that  Diodorus's  idea  of  'for  all  times'  is  basic  for 
the  Scholastics  too.  They  conducted  a  complicated  discussion  which 
shows  that  the  range  of  problems  considered  was  much  wider  than 
might  be  expected  from  what  we  have  said  here  (cf.  30.17 f.). 

A  noteworthy  exception  to  the  premise  that  a  consequence  is  a 
proposition*  is  found  in  Burleigh: 

30.11  It  is  also  to  be  noted  that  the  (contradictory)  opposite 
of  the  antecedent  does  not  follow  from  the  opposite  of  the 
consequent  in  every  valid  consequence,  but  only  in  non- 
syllogistic  consequences.  For  in  syllogistic  consequences  the 
antecedent  has  no  opposite,  because  a  syllogistic  antecedent 
is  an  unconnected  plurality  of  propositions  (propositio  plures 
inconiuncle)  and  because  such  an  antecedent  has  no  opposite 
at  all,  it  not  being  a  proposition  that  is  either  simply  or 
conjunctively  one.  But  in  a  syllogistic  consequence  the 
opposite  of  one  premiss  follows  from  the  opposite  of  the 
conclusion  with  the  other  premiss.  And  if  from  the  opposite  of 
the  conclusion  with  one  or  other  of  the  premisses  there 
follows  the  opposite  of  the  remaining  premiss,  then  the  original 
syllogism  was  valid.  For  that  is  how  the  Philosopher  proves 
his  syllogisms,  viz.  arguing  from  the  opposite  of  the  con- 
clusion with  one  of  the  premisses,  as  can  be  seen  in  the  first 
book  of  the  Prior  Analytics. 

'Propositio  plures  inconiuncle'  means  here,  as  usually  among  the 
Scholastics  (cf.  35.45)  not  a  compound  proposition,  not^therefore  a 
product  of  propositions,  but  a  number  of  juxtaposed  propositions. 
It  follows  that  syllogisms,  and  so  'syllogistic  consequences'  are 
not  propositions,  and  further  that  consequences  were  not  always 
thought  of  as  conditional  propositions. 


Here  again  we  begin  with  Pseudo-Scotus : 

30.12  Consequences  are  divided  thus:  some  are  material, 
others  are  formal.  A  formal  consequence  is  one  which  holds 
in  all  terms,  given  similar  mutual  arrangement  (disposilio) 
and  form  of  the  terms.  ...  A  material  consequence  is  one 

*  Prof.  L.  Minio-Paluello  tells  us  in  connection  with  Cod.  Orleans  '266,  fol.  78 
that  this  was  already  debated  in  the  middle  of  the  12th  century. 



which  does  not  hold  in  all  terms  given  similar  mutual  arrange- 
ment and  form  so  that  the  only  variation  is  in  the  terms 
themselves.  And  such  a  consequence  is  twofold :  one  is  simply 
true,  the  other  true  for  the  present  (ut  nunc).  A  simply  true 
consequence  is  one  reducible  to  a  formal  consequence  by  the 
addition  of  a  necessary  proposition.  A  correct  material 
consequence  true  for  the  present  is  one  which  is  reducible  to 
a  formal  consequence  by  the  addition  of  a  true  contingent 

So  there  are  three  kinds  of  consequence:  (1)  formal,  (2)  simple 
material,  (3)  material  ut  nunc.  The  last  two  are  reduced  to  the  first, 
but  by  means  of  different  kinds  of  proposition.  For  (2)  there  is  requir- 
ed a  necessary,  and  so  always  true,  proposition,  for  (3)  one  must 
use  a  contingent  proposition,  one  which  is  therefore  true  only  at  a 
certain  time.  An  example  of  the  reduction  of  (2)  to  (1)  is:  'A  man 
runs,  therefore  an  animal  runs'  is  reduced  to  a  consequence  of  kind 
(1)  by  means  of  the  proposition  'every  man  is  an  animal'  when  it  is 
said:  'every  man  is  an  animal,  a  man  runs,  therefore  an  animal 
runs'.  The  newly  introduced  proposition  is  necessary,  and  so  always 
true,  hence  the  consequence  reduced  by  its  means  to  (1)  is  'simply' 
valid,  valid  for  all  time. 

Another  definition  is  to  be  found  in  Ockham,  along  with  a  further 
division  of  formal  consequence: 

30.13  Of  consequences,  one  kind  is  formal,  another  material. 
Formal  consequence  is  twofold,  since  one  holds  by  an  extrinsic 
medium  concerning  the  form  of  the  proposition,  such  as 
these  rules:  'from  an  exclusive  to  a  universal  (proposition) 
with  the  terms  interchanged  is  a  correct  consequence',  'from 
a  necessary  major  and  an  assertoric  minor  (premiss)  there 
follows  a  necessary  (conclusion)'  etc.  The  other  kind  holds 
directly  through  an  intrinsic  medium  and  indirectly  through 
an  extrinsic  one  concerning  the  general  conditions  of  the 
proposition,  not  its  truth,  falsity,  necessity  or  impossibility. 
Of  this  kind  is  the  following:  'Socrates  does  not  run,  therefore 
some  man  does  not  run'.  The  consequence  is  called  'material' 
since  it  holds  precisely  in  virtue  of  the  terms,  not  in  virtue 
of  some  extrinsic  medium  concerning  the  general  conditions 
of  the  proposition.  Such  are  the  following:  'If  a  man  runs, 
God  exists',  'man  is  an  ass,  therefore  God  does  not  exist'  etc. 

This  text  is  most  important,  since  Ockham  here  introduces  a 
doctrine  analogous  to  that  of  Whitehead  and  Russell  in  their  distinc- 
tion of  formal  and  material  implication  (44.11  ff.),  analogous  only, 



because  the  basic  idea  of  implication  is  here  Diodorean  (20.08) 
instead  of  Philonian  (20.07).  Formal  implications  in  this  sense  are 
further  divided  into  two  classes  according  as  they  hold  in  virtue  of 
their  component  symbols  or  other  propositions  of  the  system. 

These  ideas  are  defined  with  some  accuracy  in  a  text  of  Albert  of 

30.14  Of  consequences,  one  kind  is  formal,  another 
material.  That  is  said  to  be  a  formal  consequence  to  which 
every  proposition  which,  if  it  were  to  be  formed,  would  be  a 
valid  consequence,  is  similar  in  form,  e.g.  lb  is  a,  therefore 
some  a  is  b\  But  a  material  consequence  is  one  such  that  not 
every  proposition  similar  in  form  to  it  is  a  valid  consequence, 
or,  as  is  commonly  said,  which  does  not  hold  in  all  terms  when 
the  form  is  kept  the  same;  e.g.  'a  man  runs,  therefore  an 
animal  runs'.  But  in  these  (other)  terms  the  consequence  is 
not  valid:  'a  man  runs,  therefore  a  log  runs'. 

We  may  compare  26.11  f.  with  this  text,  so  far  as  concerns  the 
notion  of  logical  form,  and  indeed  the  former  follows  immediately 
on  the  latter. 

Reverting  to  the  distinction  of  simple  and  ut  nunc  consequences 
(30.12)  with  a  similar  reference  to  time-variables  as  in  Diodorus 
Cronus  (20.08,  cf.  19.23),  Burleigh  formulates  this  idea  explicitly 
and  accurately: 

30.15  Of  consequences,  some  are  simple,  some  ut  nunc. 
Simple  are  those  which  hold  for  every  time,  as:  'a  man  runs, 
therefore  an  animal  runs'.  Consequences  ut  nunc  hold  for  a 
determinate  time  and  not  always,  as:  'every  man  runs, 
therefore  Socrates  runs' ;  for  that  consequence  does  not  hold 
always,  but  only  so  long  as  there  is  a  man  Socrates. 

The  first  rule  of  consequence  is  this :  in  every  valid  simple 
consequence  the  antecedent  cannot  be  true  without  the  con- 
sequent. And  so,  if  in  any  possible  given  case  the  antecedent 
could  be  true  without  the  consequent,  the  consequence 
would  not  be  valid.  But  in  a  consequence  ut  nunc,  the  ante- 
cedent ut  nunc,  i.e.  for  the  (given)  time  for  which  the  conse- 
quence holds,  cannot  be  true  without  the  consequent. 

Buridan  has  a  text  on  this  subject,  in  which  occurs  the  new  idea 
of  consequence  for  such  and  such  a  time  (ut  tunc). 

30.16  Of  material  consequences  some  are  said  to  be 
consequences  simply,  since  they  are  consequences  without 
qualification,  it  being  impossible  for  their  antecedents  to  be 
true  without  their  consequents.  .  .  .  And  others  are  said  to  be 



consequences  ut  nunc,  since  they  are  not  valid  without 
qualification,  it  being  possible  for  their  antecedents  to  be  true 
without  their  consequents.  However,  they  are  valid  ut  nunc, 
since  things  being  exactly  the  same  as  they  now  are,  it  is 
impossible  for  the  antecedent  to  be  true  without  the  conse- 
quent. And  people  often  use  these  consequences  in  ordinary 
language  (utuntur  saepe  vulgares),  as  when  we  say:  'the  white 
Cardinal  has  been  elected  Pope'  and  conclude:  'therefore  a 
Master  in  Theology  has  been  elected  Pope' ;  and  as  when  I 
say:  'I  see  such  and  such  a  man'  .  .  .  you  conclude  'therefore 
you  certainly  see  a  false  man'.  But  this  consequence  is  reduced 
to  a  formal  one  by  the  addition  of  a  true,  but  not  necessary, 
proposition,  or  of  several  true  and  not  necessary  ones,  as  in 
the  examples  given,  since  the  white  Cardinal  is  a  Master  in 
Theology  and  since  such  and  such  a  man  is  a  false  man.  In 
that  way  the  following  is  a  valid  consequence:  under  the 
hypothesis  that  there  are  no  men  but  Socrates,  Plato  and 
Robert,  'Socrates  runs,  Plato  runs  and  Robert  runs;  therefore 
every  man  runs'.  For  this  consequence  is  perfected  by  this 
true  (proposition):  'every  man  is  Socrates,  Plato  or  Robert'. 
And  it  is  to  be  known  that  to  this  kind  of  consequences  ut 
nunc  belong  permissive  consequences,  e.g.  'Plato  says  to 
Socrates:  if  you  come  to  me  I  will  give  you  a  horse'.  The 
proposition  may  be  a  genuine  consequence,  or  it  may  be  a 
false  proposition  and  no  consequence,  since  ( 1 )  if  the  antecedent 
is  impossible,  viz.  because  Socrates  cannot  come  to  Plato, 
then  the  consequence  is  simply  speaking  a  genuine  conse- 
quence, because  from  the  impossible  anything  follows  as  will 
be  said  below.  But  if  (2)  the  antecedent  is  false  but  not 
impossible,  then  the  consequence  is  valid  ut  nunc,  because 
from  whatever  is  false  anything  follows,  as  will  be  said  later, 
provided,  however,  that  we  restrict  the  name  'consequence 
ut  nunc'  to  consequences  ut  tunc,  whether  concerning  the 
past,  future,  or  any  other  determinate  time.  But  if  the  ante- 
cedent is  true,  so  that  Socrates  will  come  to  Plato,  then 
perhaps  we  should  say  that  it  is  still  a  genuine  consequence 
because  it  can  be  made  formal  by  the  apposition*  of  true 
(propositions),  when  one  knows  whatever  Plato  wills  to  do  in 
the  future,  that  his  wish  will  persist  and  that  he  will  be  able 
to  carry  it  out;  and  when  all  circumstances  are  taken  account 

*    Reading  appositas  for  oppositas. 



of  according  to  which  he  wills  it,  and  he  suffers  no  hindrance, 
so  that  he  will  be  able  to  and  will  do  what  and  when  he  wills; 
if  you  then  modify  this  proposition  so  that  it  is  true  according 
to  the  ninth  book  of  the  Metaphysics,  i.e.  'Plato  wills  to  give 
Socrates  a  horse  when  he  comes  to  him;  therefore  Plato  will 
give  Socrates  a  horse'.  If  then  these  propositions  about  Plato's 
will  and  power  are  true,  then  Plato  uttered  a  genuine  conse- 
quence ui  nunc  to  Socrates,  but  if  they  are  not  true  he  told 
Socrates  a  lie. 


If  the  rooks  and  the  crows  cawed  about  the  meaning  of  implication 
in  the  2nd  century  b.  c.  (20.06),  this  occupation  was  surely  intensi- 
fied in  the  15th  century.  For  while  the  Megarian-Stoic  school  has 
bequeathed  to  us  only  four  interpretations,  Paul  of  Venice  tells  us 
of  ten.  Not  all  his  definitions  are  comprehensible  to  us  today,  but 
perhaps  this  is  due  to  textual  corruptions.  However,  for  the  sake  of 
completeness  we  give  the  whole  list. 

30.17  Some  have  said  that  for  the  truth  of  a  conditional 
is  required  that  the  antecedent  cannot  be  true  without  the 
consequent.  .  .  . 

Others  have  said  that  for  the  truth  of  a  conditional  it  is 
not  required  that  the  antecedent  cannot  be  true  without 
the  consequent  in  the  divided  sense,  but  it  is  required  that  it 
is  not  possible  for  the  antecedent  to  be  true  without  the 
consequent  being  true. 

Thirdly  people  have  said  that  for  the  truth  of  a  conditional 
it  is  required  that  it  is  not  possible  that  the  antecedent  of 
that  consequence  be  true  unless  the  consequent  be  true  .  .  . 

Fourthly  people  have  said  that  for  the  truth  of  a  conditional 
it  is  required  that  it  is  not  possible  that  the  antecedent  be 
true  while  the  consequent  of  that  same  antecedent  is  false 
without  a  fresh  interpretation  (impositio)  .... 

Fifthly  people  say  that  for  the  truth  of  a  conditional  it  is 
required  that  if  things  are  (ita  est)  as  is  signifiable  by  the 
antecedent,  necessarily  things  are  as  is  signifiable  by  the 
consequent.  .  .  . 

Sixthly  people  say  that  for  the  truth  of  a  conditional  it  is 
required  that  it  be  not  possible  that  things  should  be  so  and 
not  so,  referring  to  the  significates  of  the  antecedent  and  of 
the  consequent*  of  that  conditional.  .  .  . 

*  Omitting  oppositi. 



Seventhly  people  say  that  for  the  truth  of  a  conditional  it 
is  required  that  it  is  not  possible  for  things  to  be  so  and  not  so, 
referring  to  the  adequate  significates  of  the  antecedent  and 
the  consequent.  .  .  . 

Eighthly  people  say  that  for  the  truth  of  a  conditional  it  is 
required  that  the  consequent  be  understood  in  the  antece- 
dent. .  .  . 

Ninthly  people  say  that  for  the  truth  of  a  condition  it  is 
required  that  the  adequate  significate  of  the  consequent  be 
understood  in  the  antecedent. 

Tenthly  people  say  that  for  the  truth  of  a  conditional  it  is 
required  that  the  opposite  of  the  consequent  be  incompatible 
with  the  antecedent.  .  .  . 

For  the  distinction  between  the  first  two  of  those  definitions  the 
following  text  of  Buridan  is  instructive.  * 

30.18  Then  there  is  the  rule  .  .  .  ,  that  the  consequence  is 
valid  when  it  is  impossible  that  things  are  as  signified  by  the 
antecedent  without  their  being  as  is  signified  by  the  conse- 
quent. And  this  rule  can  be  understood  in  two  ways. 

In  one  way  so  that  it  would  be  a  proposition  concerning 
impossibility  in  the  composite  sense  (the  way  in  which  it  is 
usually  intended)  and  the  meaning  then  is  that  a  consequence 
is  valid  when  the  following  is  impossible:  'If  it  is  formed, 
things  are  as  is  signified  by  the  antecedent,  and  are  not  as 
is  signified  by  the  consequent.'  But  this  rule  is  invalid,  since 
it  justifies  the  fallacy:  'No  proposition  is  negative,  therefore 
some  propositions  are  negative.' 

In  the  other  way,  so  that  it  would  be  a  proposition  concern- 
ing impossibility  in  the  divided  sense,  so  that  the  meaning  is: 
a  consequence  is  valid  when  whatever  is  stated  in  the  ante- 
cedent cannot  possibly  be  so  without  whatever  is  stated 
in  the  consequent  being  so.  And  it  is  clear  that  this  rule  would 
not  prove  the  fallacy  true;  for  whatever  'no  proposition  is 
negative'  states,  is  possibly  so,  although  things  are  not  as  the 
other  (proposition  of  the  fallacy)  states;  for  if  they  were, 
affirmatives  would  persist  but  all  negatives  would  be  annihi- 

*   This  text  was  kindly  communicated  by  Prof.  E.  Moody. 



The  notion  of  implication  is  closely  connected  with  that  of  dis- 
junction. Hence  two  characteristic  texts  are  in  place  to  illustrate 
the  problems  connected  with  the  latter.  Peter  of  Spain  writes: 

30.19  For  the  truth  of  a  disjunctive  (proposition)  it  is 
required  that  one  part  be  true,  as  'man  is  an  animal  or  crow  is 
a  stone',  and  it  is  allowed  that  both  of  its  parts  be  true,  but 
not  so  properly,  as  'man  is  an  animal  or  horses  can  whinny'. 
For  its  falsity  it  is  required  that  both  of  its  parts  be  false,  as 
'man  is  not  an  animal  or  horse  is  a  stone'. 

Peter's  idea  of  disjunction  is  evidently  rather  hazy,  for  he  wavers 
between  the  exclusive  (20.14)  and  the  non-exclusive  (20.18)  dis- 
junction, describing  the  latter  as  'less  proper'  though  at  the  same 
time  determining  falsity  in  a  way  suitable  to  it  alone.  Which  of  the 
two  is  'proper'  must  have  been  debated  even  in  the  14th  century, 
as  can  be  seen  from  Burleigh's  fine  text: 

30.20  Some  say  that  for  the  truth  of  a  disjunctive  it  is 
always  required  that  one  part  be  false,  because  if  both  parts 
were  false  it  would  not  be  a  true  disjunctive;  for  disjunction 
does  not  allow  those  things  which  it  disjoins  to  be  together, 
as  Boethius  says.  But  I  do  not  like  that.  Indeed  I  say  that 
if  both  parts  of  a  disjunctive  are  true,  the  whole  disjunctive 
is  true.  And  I  prove  it  thus.  If  both  parts  of  a  disjunctive  are 
true,  one  part  is  true;  and  if  one  part  is  true,  the  disjunctive 
is  true.  Therefore  (arguing)  from  the  first  to  the  last:  if  both 
parts  of  a  disjunctive  are  true,  the  disjunctive  is  true. 

Further,  a  disjunctive  follows  from  each  of  its  parts,  but  it 
is  an  infallible  rule  that  if  the  antecedent  is  true,  the  conse- 
quent is  true;  therefore  if  each  part  is  true  the  disjunctive  is 

I  say  therefore,  that  for  the  truth  of  a  disjunctive  it  is  not 
required  that  one  part  be  false. 

Burleigh  therefore  definitely  sides  with  those  who  understand 
disjunction  as  non-exclusive.  Also  to  be  remarked  in  this  text  are 
the  two  propositional  consequences  formulated  with  exemplary 

30.201  If  A  and  B,  then  A. 

30.202  If  A,  then  A  or  B. 



The  Scholastics  made  no  explicit  difference  between  conse- 
quences pertaining  to  propositional  and  to  term-logic.  Yet  they 
usually,  at  least  after  Ockham,  dealt  with  the  former  first.  It  is 
convenient  in  this  connection  to  quote  a  text  from  Paul  of  Venice  in 
which  he  collects  the  terminology  used  of  so-called  hypothetical 
propositions.  After  that  we  give  three  series  of  texts,  one  from 
Kilwardby  (first  half  of  the  13th  century),  one  from  Albert  of  Saxony 
(second  half  of  the  13th  century)  and  the  third  from  Paul  of  Venice 
(first  half  of  the  15th  century).  To  those  we  add  some  texts  from 
Buridan  about  consequences  ut  nunc.  We  cannot  claim  to  survey 
even  the  essentials  of  scholastic  propositional  logic,  for  this  is  as 
yet  too  little  explored.  The  texts  cited  serve  only  as  examples  of 
the  problems  considered  and  the  methods  applied. 


31.01  Some  posit  five  kinds  of  hypotheticals,  some  six, 
others  seven,  others  ten,  others  fourteen  etc.  But  leaving  all 
those  aside,  I  say  that  there  are  three  and  no  more  kinds  of 
hypotheticals  that  do  not  coincide  in  significance,  viz.  the 
copulative,  disjunctive,  and  conditional  to  which  the  rational 
is  to  be  counted  equivalent.  For  I  do  not  see  that  the  temporal, 
local  and  causal  are  hypothetical,  still  less  those  formed  and 
constituted  by  other  adverbial  and  connective  particles.  These 
are  only  hypothetical  by  similitude,  e.g.  'I  have  written  as  you 
wanted',  'Michael  answers  as  I  tell  him'.  Similarly  the  com- 
parative, e.g.  'Socrates  is  as  good  as  Plato',  'Socrates  is  whiter 
than  Plato'.  Again,  the  relative,  e.g.  'I  see  a  man  such  as 
you  see'.  .  .  .  Similarly  the  inhibitive,  e.g.  'Socrates  takes 
care  than  no-one  confute  him'.  Again  the  elective,  e.g.  'it  is 
better  to  concede  that  your  reply  is  bad  than  to  concede 
something  worse'.  Similarly  the  subjunctive,  e.g.  'I  saw  to  it 
that  you  answered  well'.  Similarly  the  expletive,  e.g.  'you 
may  be  moving  but  you  are  not  running!'  Thus  by  taking* 
the  other  particles  in  turn  one  can  form  a  very  great  number 
of  (pretended)  hypotheticals. 


We  take  a  first  series  of  consequences  from  Kilwardby's  com- 
mentary on  the  Prior  Analytics  of  Aristotle.  Kilwardby  does  not 

*    Reading  discurrendo  for  distribuendo. 



always  distinguish  very  clearly  between  propositional  and  term- 
relationships  (cf.  30.04),  so  that  'antecedent'  and  'consequent'  must 
sometimes  be  understood  as  referring  to  the  subject  and  predicate 
of  universal  affirmative  propositions. 

31.02  What  is  understood  in  some  thing  or  things,  follows 
from  it  or  from  them  by  a  necessary  and  natural  consequence ; 
and  so  of  necessity  if  one  of  a  pair  of  opposites  is  repugnant  to 
the  premisses  (of  a  syllogism)  the  other  follows  from  them. 

31.03  If  one  of  the  opposites  does  not  follow,  the  other 
can  stand. 

31.04  If  one  of  the  opposites  stands,  the  other  cannot. 

31.05  What  does  not  follow  from  the  antecedent  does 
not  follow  from  the  consequent. 

31.06  What  follows  from  the  consequent  follows  from  the 

31.07  What  is  compossible  with  one  of  two  equivalents 
(convertib ilium)  is  so  with  the  other. 

31.08  It  is  to  be  said  that  a  negation  can  be  negated,  and 
so  there  is  a  negation  of  negation,  but  this  second  negation 
is  really  an  affirmation,  though  accidentally  (secundum  quid) 
and  vocally  a  negation.  For  a  negation  which  supervenes  on 
a  negation  destroys  it,  and  in  destroying  it  posits  an  affirma- 

31.09  If  there  necessarily  follows  from  'A  is  white'  lB  is 
large',  then  from  the  denial  (destrudio)  of  the  consequent:  if 
B  is  not  large,  A  is  not  white. 

31.10  A  disjunctive  follows  from  each  of  its  parts,  and  by  a 
natural  consequence;  for  it  follows:  if  you  sit,  then  you  sit 
or  you  do  not  sit. 

31.11  If  the  antecedent  is  contingent  or  possible,  so  is  the 

31.12  It  is  not  necessary  that  what  follows  from  the  ante- 
cedent follows  from  the  consequent. 


Secondly  we  give  a  series  of  texts  from  the  Peruiilis  Logica  of 
Albert  of  Saxony  in  which  the  doctrine  of  consequences  can  be  seen 
in  a  highly  developed  state.  Albert  is  here  so  closely  dependent  on 
Buridan  that  he  often  simply  copies  him.  But  there  is  much  that  he 
formulates  more  clearly,  and  the  available  text  of  Buridan  is  not  so 
good  as  that  of  the  Peruiilis  Logica.   Buridan  himself  is  not  the 



original  author  of  his  doctrine  of  consequences;  much  of  it  comes 
from  Ockham,  and  some  even  from  Peter  of  Spain. 

As  in  this  whole  section,  the  contemporary  range  of  problems  is 
only  barely  illustrated. 

Albert's  definitions  of  antecedent  and  consequent  deserve  to  be 
quoted  first: 

31.13  That  proposition  is  said  to  be  antecedent  to  another 
which  is  so  related  to  it  that  it  is  impossible  that  things  be 
as  is  signifiable  by  it  without  their  being  as  is  in  any  way 
signifiable  by  the  other,  keeping  fixed  the  use  (impositio)  of 
the  terms. 

Like  all  Scholastics  of  the  14th  century  and  after,  Albert  makes  a 
clear  distinction  between  a  rule  of  consequence  and  the  consequence 
itself.  A  rule  is  a  metalogical  (more  exactly  a  meta-metalogical) 
description  of  the  form  of  a  valid  consequence.  The  consequence 
itself  is  a  proposition  having  this  form.  That  generally  holds  good; 
but  some  of  Albert's  rules  are  conceived  as  propositional  forms  like 
the  Stoic  inference-schemata  (21.22)  -  cf.  the  fifth  (31.18)  -  only 
with  this  difference,  that  the  variables  are  here  evidently  metalo- 
gical, i.e.  to  be  substituted  with  names  of  propositions,  not  with 
propositions  themselves  as  is  the  case  with  the  Stoic  formulae. 

31.14  The  first  (rule  of  simple  consequence)  is  this:  from  an 
impossible  proposition  every  other  follows.  Proof:  from  the 
nominal  definitions  of  antecedent  and  consequent  given  in  the 
first  chapter.  For  if  a  proposition  is  impossible,  it  is  impossible 
that  things  are  as  it  indicates,  and  are  not  as  any  other 
indicates ;  therefore  the  impossible  proposition  is  antecedent  to 
every  other  proposition,  and  hence  every  proposition  follows 
from  an  impossible  one.  This  it  is  which  is  usually  expressed: 
anything  follows  from  the  impossible.  And  so  it  follows:  man 
is  an  ass,  therefore  a  man  runs;  since  the  antecedent  being 
impossible,  if  things  are  not  as  the  consequent  indicates,  it  is 
impossible  that  they  should  be  as  the  antecedent  indi- 

31.15  Second  rule:  A  necessary  proposition  follows  from 
any  proposition.  This  is  again  proved  by  the  nominal  defini- 
tions of  antecedent  and  consequent.  For  it  is  impossible  that 
things  should  not  be  as  a  necessary  proposition  indicates, 
if  they  are  as  any  other  (proposition)  indicates.  Hence  a 
necessary  proposition  is  a  consequent  of  any  proposition.  It 
follows  therefore  that  this  consequence  is  valid:  'a  man  runs, 
therefore  God  exists',  or  '(therefore)  ass  is  an  animal'. 



The  proofs  of  these  two  rules  are  very  typical  of  the  Scholastic 
approach  to  propositional  logic  and  show  how  different  it  is  both  to 
that  of  the  Megarian-Stoics  and  that  of  the  moderns.  The  essential 

scholastic  point  is  that  a  consequence  does  not  unite  two  states  of 
affairs  but  two  propositions  (in  the  scholastic  sense,  which  includes 
the  mental  propositions,  cf.  26.03).  Let  'P'  be  the  name  of  the 
proposition  expressing  the  state  of  affairs  p,  and  lQ'  the  name  of 
the  proposition  expressing  the  state  of  affairs  q,  the  proof  of  the 
first  consequence  can  be  presented  thus: 

As  axiom  is  presupposed 

31.151  If  p  cannot  be  the  case  then  (p  and  q)  cannot  be  the 

Then  the  process  is: 

(1)  P  is  impossible  (hypothesis) 

(2)  p  cannot  be  the  case  (by  (1)  and  the  definition  of 


(3)  (p  and  not  q)  cannot  be  the  case      (by  (2)  and  31.151) 

(4)  Q  is  the  consequent  of  P  (by  the  definition  and  (3)) 

(5)  Q  follows  from  P  (by  definition) 
And  this  was  to  be  proved,  Q  being  any  proposition. 

Thus  we  can  see  that  a  metalogical  thesis  about  a  relationship 
(consequence)  between  propositions  is  proved  through  reduction  to 
logical  laws  concerning  relationships  between  states  of  affairs. 

31.16  Third  rule:  (1)  From  any  proposition  there  follows 
every  other  whose  contradictory  opposite  is  incompatible 
with  it  (the  first).  And  (2)  from  no  proposition  does  there 
follow  another  whose  contradictory  opposite  is  compatible 
with  it,  where  (the  expression)  'a  proposition  is  compatible 
with  another'  is  to  be  understood  in  the  sense  that  the  state  of 
affairs  (sic  esse)  which  the  one  indicates  is  compatible  with  that 
which  the  other  indicates.  .  .  . 

The  first  part  of  the  rule  is  proved  (thus) :  Let  us  suppose 
that  the  proposition  B  is  incompatible  with  the  proposition  A. 
I  say  (then)  that  from  A  there  follows  the  contradictory 
opposite  of  B,  i.e.  not-B.  This  is  evident,  for  A  and  B  are 
incompatible  and  therefore  (either  A)  is  impossible,  so  that 
every  proposition  follows  from  it,  by  the  first  rule;  or  A  is 
possible,  then  necessarily  if  A  is  the  case,  either  B  or  not-B  is 
the  case,  since  one  part  of  a  pair  of  contradictory  opposites  is 
always  true.  But  it  is  impossible  that  if  A  is  the  case,  B  is  the 



case,  by  hypothesis.  Therefore  it  is  necessary  that  if  A  is  the 
case,  not-B  is  the  case.  Therefore  not-B  follows  from  A. 

The  second  part  of  the  rule  is  proved  (thus) :  if  A  and 
not-B  are  true  together,  then  this  holds:  If  A  is  the  case, 
B  is  not  the  case.  But  since  B  and  not-B  are  not  true  together, 
it  is  possible  that  if  A  is  the  case,  B  is  not  the  case.  Therefore 
B  does  not  follow  from  A. 

31.17  Fourth  rule:  for  every  valid  consequence,  from  the 
contradictory  opposite  of  the  consequent  there  follows  the 
contradictory  opposite  of  the  antecedent.  This  is  evident, 
since  on  the  supposition  that  B  follows  from  A,  I  say  that 
not-^4  follows  from  not-B.  For  either  it  is  so,  or  it  is  possible 
that  A  and  not-B  are  true  together,  by  the  previous  rule.  But 
it  is  necessary  that  if  A  is  the  case,  B  is  the  case.  Therefore  B 
and  not-B  will  be  true  together,  which  is  impossible,  by  the 
accepted  (communis)  principle  'it  is  impossible  that  two  contra- 
dictories should  be  true  together'.  .  .  . 

31.18  Fifth  rule:  if  B  follows  from  A,  and  C  from  B,  then  (1) 
C  follows  from  A ;  and  (2)  C  follows  from  everything  from 
which  B  follows;  and  (3)  what  does  not  follow  from  A,  does 
not  follow  from  B;  and  (4)  from  everything  from  which  C  does 
not  follow,  B  too  does  not  follow.  That  is  to  say,  in  current 
terms,  all  the  (following)  consequences  are  valid:  (1)  Whatever 
follows  from  the  consequent  follows  from  the  antecedent; 
(2)  The  consequent  of  this  consequence  follows  from  all  that 
from  which  the  antecedent  follows;  (3)  What  does  not  follow 
from  the  antecedent  does  not  follow  from  the  consequent; 
(4)  The  antecedent  does  not  follow  from  that  from  which  the 
consequent  does  not  follow.  This  rule  has  four  parts. 

The  first  (part)  is:  If  B  follows  from  A,  and  C  from  B,  then 
C  follows  from  A.  For  on  the  supposition  that  B  follows  from 
A,  if  things  are  as  A  indicates,  they  are  also  as  B  indicates,  by 
the  nominal  definition  of  antecedent  and  consequent.  And 
on  the  supposition  that  C  follows  from  B,  if  things  are  as  B 
indicates,  they  are  also  as  C  indicates.  Therefore,  if  things 
are  as  A  indicates,  they  are  also  as  C  indicates.  And  accordingly 
C  follows  from  A. 

The  second  part  is  evident,  since  nothing  from  which  B 
follows  can  be  the  case  if  B  is  not  the  case ;  and  as  B  cannot  be 
the  case  if  C  is  not  the  case,  it  follows  also:  C  follows  from  all 
from  which  B  follows.  And  by  'being  the  case'  is  to  be  under- 
stood :  being  as  B  indicates  .... 



31.19  Sixth  rule:  (1)  It  is  impossible  that  false  follows  from 
true.  (2)  It  is  also  impossible  that  from  possible  follows 
impossible.  (3)  It  is  also  impossible  that  a  not  necessary 
proposition  follows  from  a  necessary  one.  (The  first  part) 
is  evident  by  the  nominal  definition  of  antecedent  and  conse- 
quent. For  if  things  are  as  the  antecedent  indicates,  they  are 
also  as  the  consequent  indicates,  and  accordingly,  when  the 
antecedent  is  true,  the  consequent  is  true  and  not  false.  The 
second  part  is  evident,  for  if  things  can  be  as  the  antecedent 
indicates,  they  can  also  be  as  the  consequent  indicates;  and 
accordingly,  when  the  antecedent  is  possible,  the  consequent 
also  (is  possible).  The  third  part  is  evident,  for  if  things 
necessarily  are  as  the  antecedent  indicates,  they  must  also 
(necessarily)  be  as  the  consequent  indicates. 

31.20  There  follows  from  this  rule:  (1)  if  the  consequent 
of  a  consequence  is  false,  its  antecedent  is  also  false ;  (2)  further, 
if  the  consequent  of  a  consequence  is  impossible,  its  antece- 
dent is  also  impossible ;  (3)  further  if  the  consequent  of  a  con- 
sequence is  not  necessary,  its  antecedent  also  is  not  necessary. 

And  I  purposely  (notanter)  say,  'if  the  consequent  is  not 
possible'  and  not  'if  the  consequent  is  not  possibly  true',  since 
in  this  (consequence) :  'every  proposition  is  affirmative, 
therefore  no  proposition  is  negative',  its  antecedent  is  possible 
and  its  consequent  too  is  possible,  but  although  it  is  possible,  it 
is  impossible  that  it  be  true,  as  was  said  above.  And  yet  true 
can  follow  from  false,  and  possible  can  follow  from  impossible, 
and  necessary  can  follow  from  not  necessary,  as  is  evident 
from  Aristotle  in  the  second  (book)  of  the  Prior  (Analytics, 
ch.  2)  (16.32) 

31.21  Seventh  rule:  if  B  follows  from  A  together  with  one 
or  more  additional  necessary  propositions,  then  B  follows 
from  A  alone.  Proof:  B  is  either  necessary  or  not  necessary.  If 
it  is  necessary,  it  follows  from  A  alone,  by  the  second  rule, 
since  the  necessary  follows  from  any  (proposition).  But  if  B 
is  not  necessary,  then  A  is  either  possible  or  impossible. 
Suppose  A  is  impossible  then  again  B  follows  from  A  alone  as 
also  from  A  with  an  additional  necessary  proposition,  by  the 
first  rule.  Since  from  the  impossible,  anything  follows.  But 
suppose  A  is  possible,  then  if  A  is  the  case  it  is  impossible  that 
B  is  not  the  case,  or,  if  A  is  the  case  it  is  possible  that  B  is 
not  the  case.  On  the  first  supposition,  B  follows  from  A  alone, 
as  also  from  A  with  an  additional  necessary  proposition,  by  the 



nominal  definition  of  antecedent  and  consequent.  But  sup- 
posing that  if  A  is  the  case  it  is  possible  that  B  is  not  the  case, 
then  if  A  is  the  case,  A  and  the  additional  necessary  pro- 
position must  be  true  together.  For  it  is  impossible  that*  A 
should  not  be  the  case,  since  it  is  not  possible  that  if  A  is  the 
case,  A  is  not  the  case.  And  accordingly,  granted  that  A  is  the 
case,  i.e.  granted  that  things  are  as  A  indicates,  it  is  necessary 
that  they  should  be  as  A  and  the  additional  necessary  pro- 
position indicate.  Therefore  from  A  there  follow  A  and  the 
additional  necessary  proposition.  And  as  B  follows  from  A 
and  the  additional  necessary  proposition,  one  obtains  the 
probandum  by  means  of  the  first  part  of  this  rule,  (viz.) 
that  B  follows  from  A  alone,  which  was  to  be  proved. 

The  rule  could  be  formulated: 

31.211  If  C  is  necessary,  then :  if  B  follows  from  A  and  C,  B 
follows  from  A  alone, 

and  the  proof  is  contained  in  the  words:  'if  A  is  the  case,  A  and  the 
additional  necessary  proposition  must  be  the  case'  and  the  subse- 
quent justification.  For  in  fact,  if  C  is  necessary,  then  if  we  have  A, 
we  have  A  and  C,  and  then  if  B  follows  from  A  and  C,  B  follows 
from  A.  The  passage  previous  to  the  words  just  quoted  is  therefore 
superfluous,  but  it  has  been  retained  as  characteristic  of  the  scho- 
lastic approach. 

31.22  Eighth  rule:  every  consequence  of  this  kind  is  for- 
mal: 'Socrates  exists,  and  Socrates  does  not  exist,  therefore  a 
stick  stands  in  the  corner'.  Proof:  By  formal  consequence  it 
follows:  'Socrates  exists  and  Socrates  does  not  exist,  therefore 
Socrates  exists',  from  a  complete  copulative  proposition  to 
one  of  its  parts.  Further  it  follows:  'Socrates  exists  and 
Socrates  does  not  exist,  therefore  Socrates  does  not  exist'  by 
the  same  rule.  And  it  further  follows:  'Socrates  exists,  there- 
fore either  Socrates  exists  or  a  stick  stands  in  the  corner'. 
The  consequence  holds,  since  from  every  categorical  propo- 
sition a  disjunctive  proposition  is  deducible  (infert)  of  which 
it  is  a  part.  And  then  again:  'Socrates  exists  and  Socrates  does 
not  exist;  therefore  (by  the  second  part  of  this  copulative 
proposition):  Socrates  does  not  exist;  therefore  a  stick  stands 
in  the  corner'.  The  consequence  holds  since  the  consequence 
is  formal  from  a  disjunctive  with  the  denial  (destrudio)  of  one 

*  Omitting  necessariam. 



of  its  parts  to  the  other.  And  so  every  proposition  similar  in 
form  to  this  would  be  a  valid  consequence  if  it  were  formed. 
This  rule  is  usually  expressed  in  the  following  words :  'from 
every  copulative  consisting  of  contradictorily  opposed  parts, 
there  follows  any  other  (proposition  by)  formal  consequence'. 

This  text  is  undoubtedly  one  of  the  peaks  of  scholastic  prepo- 
sitional logic.  Both  the  rule  and  its  proof  were  part  of  the  scholastic 
capital.  It  is  to  be  found  in  Pseudo-Scotus  in  the  form: 

31.23  From  every  proposition  evidently  implying  a  con- 
tradiction, any  other  formally  follows.  So  there  follows  for 
instance:  'Socrates  runs  and  Socrates  does  not  run,  therefore 
you  are  at  Rome.' 

The  proof  in  31.22  relies  on  the  following  laws  as  axioms,  which 
are  expressly  formulated : 

31.221  If  P  and  Q  then  P. 

31.222  If  P  and  Q  then  Q. 

31.223  If  P  then,  P  or  Q. 

31.224  If  P  or  Q,  then,  if  not-P  then  Q. 

The  proof  runs  thus: 

(1)  P  and  not-P  (hypothesis) 

(2)  P  (by  (1)  and  31.221  with  sub- 

stitution  of   'not-P'   for   lQ') 

(3)  P  or  Q  (by  (2)  and  31.223  (cf.  31.10)) 

(4)  not-P  (by  (1)  and  31.222  with  sub- 

stitution  of   'not-P'    for   lQ') 

(5)  Q  (by  (3),  (4)  and  31.224) 
And  this  was  the  probandum,  Q  being  any  proposition  at  all. 

Of  the  laws  used  in  this  proof,  31.221-2  are  to  be  found  in  Ockham 
(31.24)  and  were  also  familiar  to  Paul  of  Venice.  31.223  is  the  modern 
law  of  the  factor,  accepted  by  the  Scholastics  from  the  time  of  Kil- 
wardby  (31.10).  31.224  is  the  later  modus  tollendo  ponens,  ana- 
logous to  the  fifth  indemonstrable  of  the  Stoics  (22.08),  but  using 
non-exclusive  disjunction. 


Next  we  give  some  rules  for  copulative  propositions,  from  Paul 
of  Venice. 

31.25  For  the  truth  of  an  affirmative  copulative  (propo- 
sition) there  is  required  and  suffices  the  truth  of  both  parts 
of  the  copulative.  .  .  . 

31.26  A  corollary  from  this  rule  is  the  second:  that  for  the 



falsity  of  an  affirmative  copulative  the  falsity  of  one  of  its  parts 
is  sufficient.  .  .  . 

31.27  The  third  rule  is  this:  for  the  possibility  of  the  copu- 
lative it  is  required  and  suffices  that  each  of  its  principal  parts 
is  possible  and  each  is  compossible  with  each  -  or  if  there  are 
more  than  two,  with  all.  .  .  . 

31.28  From  this  follows  the  fourth,  viz.:  for  the  impos- 
sibility of  a  copulative  it  is  sufficient  and  requisite  that  one  of 
its  principal  parts  be  impossible  or  that  one  be  not  compossible 
with  the  other,  or  the  others.  .  .  . 

31.29  The  fifth  rule  is  this:  for  the  necessity  of  an  affir- 
mative copulative,  the  necessity  of  every  one  of  its  parts.  .  .  . 

31.30  From  this  rule  follows  the  sixth:  that  for  the  contin- 
gence  of  a  copulative  it  is  required  and  suffices  that  one  of  its 
categorical  principal  parts  be  contingent  and  compossible  with 
the  other,  or  with  all  others  if  there  are  more  than  two. 

Similar  rules  for  'known',  'known  as  true',  'credible'  follow. 
For   disjunctives,   the   same   author   gives   the   following   rules, 
among  others: 

31.31  From  what  has  been  said  (cf.  31.223)  there  follow 
four  corollaries.  The  first  is:  if  there  is  an  affirmative  dis- 
junctive .  .  .  composed  of  two  categoricals  of  which  one  is 
superordinate  to  the  other  by  reason  of  a  term  or  terms  in  it, 
the  argument  is  valid  to  the  superordinate  part;  it  follows 
e.g.:  you  run  or  you  are  in  motion,  therefore  you  are  in 
motion.  .  .  . 

31.32  The  second  corollary  is  this:  if  there  is  a  disjunctive 
consisting  of  two  categoricals  of  which  one  is  possible,  the 
other  impossible,  the  argument  to  the  possible  part  is  valid. 
Hence  it  follows  validly:  'there  is  no  God,  or  you  do  not  exist, 
therefore  you  do  not  exist';  'you  are  an  ass  or  you  run, 
therefore  you  run'. 

31.33  The  third  corollary  is  this:  if  there  be  formed  a 
disjunctive  of  two  categoricals  that  are  equivalent  (convertibi- 
libus),  the  argument  to  each  of  them  is  valid,  for  it  follows 
validly:  'there  is  no  God  or  man  is  an  ass,  therefore  man  is  an 
ass'.  And  from  the  same  antecedent  it  follows  that  there  is  no 
God,  since  those  categoricals,  being  impossible,  are  equivalent. 
It  further  follows:  'you  are  a  man  or  you  are  risible,  therefore 
you  are  risible',  and  it  also  follows  that  you  are  a  man. 

31.34  The    fourth   corollary  is   this:   if   a    disjunctive    be 



formed  of  two  categoricals  of  which  one  is  necessary  and  the 
other  contingent,  the  argument  is  valid  to  the  necessary  part. 
Hence  it  follows  validly:  'you  run  or  God  exists,  therefore 
God  exists'.  And  it  is  not  strange  that  all  such  consequences 
hold,  for  the  consequent  follows  of  itself  immediately  (con- 
tinue) from  each  part  of  the  disjunctive,  hence  it  must  follow 
too  from  the  disjunctives  themselves. 

31.35  The  eighth  principal  rule  is  this:  from  an  affirmative 
disjunctive  ...  to  the  negative  copulative  formed  of  the 
contradictories  of  the  parts  of  the  disjunctive  is  a  valid 
argument.  The  proof  is  that  the  affirmative  copulative  formed 
of  the  contradictories  of  the  parts  of  the  disjunctive  contradicts 
the  disjunctive,  therefore  the  contradictory  of  that  copulative, 
formed  by  prefixing  a  negative,  follows  from  the  disjunctive. 
For  example  'you  run  or  you  are  in  motion,  therefore:  not, 
you  do  not  run  and  you  are  not  in  motion' ;  'God  exists  or  no 
man  is  an  ass,  therefore:  not,  there  is  no  God  and  man  is  an 
ass'.  Those  consequences  are  evident,  since  the  opposites 
of  the  consequents  are  incompatible  with  their  antecedents, 
as  has  been  said. 

31.36  From  that  rule  there  follows  as  a  corollary  that  from 
an  affirmative  copulative  ...  to  a  negative  disjunctive 
formed  of  the  contradictories  of  the  parts  of  the  copulative  is 
a  valid  argument.  Hence  it  follows  validly:  'you  are  a  man 
and  you  are  an  animal,  therefore:  not,  you  are  not  a  man  or 
you  are  not  an  animal'.  Similarly  it  follows:  'you  are  not  a 
goat  and  you  are  not  an  ass,  therefore :  not*,  you  are  a  goat  or 
you  are  an  ass'.  .  .  . 

The  last  two  texts  contain  two  of  the  so-called  'de  Morgan '  laws. 
So  far  as  is  known,  they  first  occur  in  Ockham  (31.37)  and  Burleigh 
(31.38)**.  The  latter  gives  them  in  the  form  of  equivalences. 

31.39  If  one  argues  from  an  affirmative  conditional, 
characterized  (denominaia)  by  'if,  to  a  disjunctive  consisting 
of  the  contradictory  of  the  antecedent  and  the  consequent* 
of  the  conditional  the  consequence  is  formal.  The  proof  is 
that  this  consequence  is  formal :  'if  you  are  a  man  you  are  an 
animal;  therefore,  you  are  not  a  man  or  you  are  an  animal'. 
And  no  one  example  is  more  cogent  than  another,  therefore  all 
are  valid  consequences. 

*   Adding  non. 
*  *   But  Peter  of  Spain  (  Traclalus  Syncategorematum)  has  the  doctrine  of  31.36. 
*  *  *  Reading  consequente  for  consequentis. 



Buridan  writes: 

31.40  And  it  is  to  be  noted  that  a  proportionate  conclusion 
is  to  be  posited  concerning  consequences  ut  nunc  (i.e.  propor- 
tionate to  that  concerning  simple  consequences),  viz.  that 
from  every  false  proposition  there  follows  every  other  by  a 
consequence  ut  nunc,  because  it  is  impossible  that  things  being 
as  they  now  are  a  proposition  which  is  true  should  not  be  true. 
And  so  it  is  not  impossible  that  it  should  be  true,  however 
anything  else  may  not  be  true.  And  when  the  talk  is  of  the 
past  or  future,  then  it  can  be  called  a  consequence  ut  nunc,  or 
however  else  you  like  to  call  it,  e.g.  it  follows  by  a  consequence 
ut  nunc  or  ut  tunc  or  even  nunc  per  tunc:  'if  Antichrist  will  not 
be  generated,  Aristotle  never  existed'.  For  though  it  be  simply 
true  that  it  is  possible  that  Antichrist  will  not  exist,  yet  it  is 
impossible  that  he  will  not  exist  when  things  are  going  to  be 
as  they  will  be ;  for  he  will  exist,  and  it  is  impossible  that  he 
will  exist  and  that  he  will  not  exist. 

We  have  here  first  of  all  the  two  classical  'paradoxical'  laws  of 
material  implication: 

31.401  If  P  is  false  then  Q  follows  from  P. 

31.402  If  P  is  true,  then  P  follows  from  Q. 

Buridan  provides  an  example  of  (substitution  in)  the  first  of 
these  laws,  putting  the  proposition  'Antichrist  will  not  be  generated' 
for  lP\  and  'Aristotle  never  existed'  for  lQ\  The  first  proposition 
is,  absolutely  speaking,  possible,  so  this  cannot  be  a  case  of  simple 
consequence  (cf.  the  first  rule,  31.14  supra),  for  that  would  require 
it  to  be  absolutely  impossible.  But  the  consequence  holds  if  taken 
ut  nunc,  since  the  proposition  'Antichrist  will  not  be  generated'  will 
in  fact  be  impossible  in  what  will  be  the  circumstances.  Hence 
we  have  impossibility  for  that  time  (ut  tunc)  and  so  a  consequence 
that  holds  for  that  time. 

This  shows  that  even  consequence  for  a  given  time  is  defined  by 
means  of  impossibility.  The  difference  between  it  and  simple  conse- 
quence consists  only  in  the  kind  of  impossibility,  absolute  (for  all 
times  and  circumstances)  in  the  case  of  simple  consequence,  con- 
ditioned in  that  of  consequence  for  a  given  time. 

But  impossibility  ut  nunc  is  defined  as  simple  non-existence,  and 
so  the  proposition  'Antichrist  will  not  be  generated'  can  be  reckoned 
as  impossible  since  Antichrist  will  in  fact  be  generated.  It  follows 
that  consequence  ut  nunc  can  be  defined  without  the  help  of  the 
modal  functor,  a  proposition  ut  nunc  being  impossible  simply  when 
it  is  false. 



Another  law  of  consequence  ut  nunc  comes  from  the  same  text  of 
Buridan : 

31.41  If  a  conclusion  follows  from  a  proposition  together 
with  one  or  more  additional  propositions,  the  same  conclusion 
follows  from  that  proposition  alone  by  a  consequence  ut  nunc. 

This  rule  is  analogous  to  the  seventh  rule  given  above  for  simple 
consequence  (31.21),  which  shovvs  that  the  whole  system  of  simple 
consequences  can  be  transformed  into  a  system  of  consequences 
ut  nunc  by  everywhere  replacing  'necessary'  by  'true'  and  'impossi- 
ble'  by  'false'  and  similar  simplifications. 

Finally  we  remark  that  Buridan,  so  far  as  is  now  known,  is  the 
only  scholastic  logician  to  develop  laws  of  consequence  ut  nunc, 
though  he  devoted  much  less  space  to  them  than  to  simple  conse- 
quences. In  Paul  of  Venice  the  subject  of  consequence  ut  nunc  seems 
to  have  dropped  out  completely. 




Contrary  to  a  widespread  opinion,  the  assertoric  syllogistic  was 
not  only  not  the  only,  it  was  not  even  the  chief  subject  of  scholastic 
logic.  The  Scholastics,  like  most  of  the  Commentators  (24.271  ff.), 
thought  of  syllogisms  as  rules  (cf.  30.11)  rather  than  conditional 
propositions.  The  domain  of  syllogisms  received  a  significant  exten- 
sion through  the  introduction  of  singular  terms  already  in  Ockham's 
time.  But  the  new  formulae  thus  derived  will  here  be  given  separately 
under  the  heading  'Other  Formulae'  (§  34)  since  they  effect  an 
essential  alteration  in  the  Aristotelian  syllogistic.  In  the  present 
section  we  shall  confine  ourselves  to  that  part  of  the  scholastic 
treatment  which  can  still  be  deemed  Aristotelian.  Here,  too, 
everything  is  treated  purely  metalogically  (except  in  some  early 
logicians  such  as  Albert  the  Great),  but  that  is  quite  in  the  Aristo- 
telian tradition  (14.25ff.). 

The  most  important  contributions  are  these:  (1)  the  devising  of 
numerous  mnemonics  for  the  syllogistic  moods  and  their  inter- 
relationships, culminating  in  the  pons  asinorum.  (2)  The  systematic 
introduction  and  thorough  investigation  of  the  fourth  figure.  (3) 
The  position  and  investigation  of  the  problem  of  the  null  class,  which 
has  already  received  mention  in  connection  with  appellation  (§  28,  B). 


L.  Minio-Paluello  has  recently  made  the  big  discovery  of  an  early 
attempt  to  construct  syllogistic  mnemonics,  in  a  MS  of  the  early 
13th  century.  This  deprives  of  its  last  claims  to  credibility  Prantl's 
legend  of  the  Byzantine  origin  of  such  mnemonics.  *  The  essentials 
are  these: 

32.06  It  is  to  be  remarked  that  there  are  certain  notations 
(notulae)  for  signifying  the  moods.  .  .  .  The  four  letters  e,  i,  o,  u 
signify  universal  affirmatives,  and  the  four  letters  /,  m,  n,  r 
signify  universal  negatives,   and  the  three**  a,  s,  t  signify 

*  Carl  Prantl,  relying  on  a  single  MS,  ascribed  to  Michael  Psellus  (1018  to 
1078/96)  the  'Lvvotyit;  elq  ttjv  'AptaTOTeXou<;  Xoyix^v  eTuarrjpgv  in  which  such 
mnemonics  occur,  and  stated  that  the  Summulae  of  Peter  of  Spain  was  a  trans- 
lation from  that  (32.01).  In  that  opinion  he  was  the  victim  of  a  great  mistake, 
since  M.  Grabmann  (32.02)  following  C.  Thurot  (32.03),  V.  Rose  (32.04)  and 
R.  Stapper  (32.05)  has  shown  the  Hvvotyic,  to  be  by  George  Scholarios  (1400-1464) 
and  a  translation  of  the  Summulae. 
*  *   Reading  '3'  for  '4'. 



particular  affirmatives,  and  b,  c,  d  signify  particular  negatives. 
So  the  moods  of  the  first  figure  are  shown  in  the  following 
verse:  uio,  non,  est  (tost),  lac,  uia,  mel,  uas,  erp,  arc.  Thus  the 
first  mood  of  the  first  figure,  signified  by  the  notation  uio, 
consists  of  a  first  universal  affirmative  and  a  subsequent 
universal  affirmative,  (and)  concludes  to  a  universal  affirma- 
tive; e.g.:  All  man  is  animal  and  all  risible  is  man,  therefore 
all  risible  is  animal.  .  .  .  The  moods  of  the  second  figure  are 
shown  in  the  following  verse:  ren,  erm,  vac* ,  obd.  .  .  .  The 
moods  of  the  third  figure  are  shown  in  the  following  verse: 
eua,  nee,  aut,  esa,  due,  nac. 

This  is  a  very  primitive  technique,  but  at  least  it  shows  that  the 
highly  developed  terminology  of  Peter  of  Spain***  had  antecedents 
in  Scholasticism  itself. 

We  cite  the  relevant  texts  from  the  Summulae  Logicales. 


32.07  After  giving  a  threefold  division  of  propositions  it  is 
to  be  known  that  there  is  a  threefold  enquiry  to  be  made  about 
them,  viz.  What?,  Of  what  kind?,  How  much?  'What?' 
enquires  about  the  nature  (substantia)  of  the  proposition, 
so  that  to  the  question  'What?'  is  to  be  answered  'categorical' 
or  'hypothetical';  to  'Of  what  kind?'  -  'affirmative'  or 
'negative';  to  'How  much?'  -  'universal',  'particular', 
'indefinite',  'singular'.  Whence  the  verse: 

Quae  ca  vel  hip,  qualis  ne  vel  aff,  un  quanta  par  in  sin. 
the  questions  being  in  Latin:  quae?,  qualis?,  quanta?,  and 
the  answers:  categorica,  hypoihetica,  affirmaiiva,  universalis, 
particularis,  infinita,  singularis. 

So  far  as  we  know  this  is  the  first  text  in  which  the  notions  of 
quality  and  quantity  occur.  The  full  set  of  technical  terms  connected 
therewith,  together  with  some  others,  appears  in  the  next  passage 
which  resumes  the  doctrine  of  conversion. 

32.08  The  conversion  of  propositions  having  both  terms 
in  common  but  with  the  order  reversed,  is  threefold;   viz. 

*    MS  has  'rachc\ 
*  *   Personal  thanks  are  due  to  Prof.  L.  Minio-Paluello  for  telling  us  of  this  MS 
and  helping  to  restore  the  text. 

*  *  *  This  can  not  have  originated  with  him.  Prof  L.  Minio-Paluello  informed 
me  on  24.  6.  55  that  he  had  found  the  word  'Fesiino'  in  a  MS  dating  at  the  latest 
from  1200. 



simple,  accidental,  and  by  contraposition.  Simple  conversion  is 
when  the  predicate  is  made  from  the  subject  and  conversely, 
the  quality  and  quantity  remaining  the  same.  And  in  this 
way  are  converted  the  universal  negative  and  the  particular 
affirmative.  .  .  .  The  universal  affirmative  is  similarly  con- 
verted when  the  terms  are  equivalent  (convertibilibus).  .  .  . 

Accidental  conversion  is  when  the  predicate  is  made  from 
the  subject  and  conversely,  the  quality  remaining  the  same, 
but  the  quantity  being  changed ;  and  in  this  way  the  universal 
negative  is  converted  into  the  particular  negative,  and  the 
universal  affirmative  into  the  particular  affirmative.  .  .  . 

The  law  of  accidental  conversion  of  the  universal  negative  is  not 
in  Aristotle. 

32.09  Conversion  by  contraposition  is  to  make  the  predi- 
cate from  the  subject  and  conversely,  quality  and  quantity 
remaining  the  same,  but  finite  terms  being  changed  to 
infinite  ones.  And  in  this  way  the  universal  affirmative  is 
converted  into  itself  and  the  particular  negative  into  itself, 
e.g.  'all  man  is  animal'  -  'all  non-animal  is  non-man';  'some 
man  is  not  stone'  -  'some  non-stone  is  not  non-man'.* 

Hence  the  verses: 

A  Affirms,  E  rEvokes**,  both  universal, 

/  affirms,  0  revOkes**,  both  in  particular. 

Simply  converts  fEel,  accidentally  EvA, 

AstO  by  contra(position) ;  and  these  are  all  the  conversions. 

The  classical  expressions  Barbara,  Celarent,  etc.  seem  to  have 
been  fairly  generally  known  about  1250.  After  describing  the 
assertoric  moods  Peter  of  Spain  introduced  them  thus: 

32.10  Hence  the  verses: 

Figure  the  first  to  every  kind***  concludes, 
The  second  only  yields  negations, 
Particulars  only  from  third  figure  moods. 
Barbara,  celarent,  darii,  ferion,  baralipton, 
Celantes,  dabitis,  fapesmo,  frisesomorum. 
Cesare.  camestres,  festino,  baroco,  darapti. 
Felapto,  disamis,  datisi,  bocardo,  ferison. 

*   Reading  non  homo. 
*  *  negat. 
*  *  *   viz.  problematis. 



32.11  In  those  four  verses  there  are  twenty-one  expressions 
(didiones)  which  so  correspond  to  the  twenty-one  moods  of  the 
three  figures  that  by  the  first  expression  is  to  be  understood  the 
first  mood,  and  by  the  second  the  second,  and  so  with  the  others. 
Hence  the  first  two  verses  correspond  to  the  moods  of  the 
first  figure,  but  the  third  to  the  moods  of  the  second  save  for 
its  last  expression.  It  is  to  be  known  therefore  that  by  these 
four  vowels,  viz.  A,  E,  1,0  set  in  the  aforesaid  verses  there 
are  understood  the  four  kinds  of  proposition.  By  the  vowel  A 
is  understood  the  universal  affirmative,  by  E  the  universal 
negative,  by  /  the  particular  affirmative,  by  0  the  particular 

32.12  Further  it  is  to  be  known  that  in  each  expression 
there  are  three  syllables  representing  three  propositions,  and 
if  there  is  anything  extra  it  is  superfluous,  excepting  M  as 
will  appear  later.  And  by  the  first  syllable  is  understood  the 
major  proposition,  similarly  by  the  second  the  second  proposi- 
tion, and  by  the  third  the  conclusion;  e.g.  the  first  expression, 
viz.  Barbara,  has  three  syllables,  in  each  of  which  A  is  set, 
and  A  set  three  times  signifies  that  the  first  mood  of  the  first 
figure  consists  of  two  universal  affirmatives  concluding  to  a 
universal  affirmative;  and  thus  it  is  to  be  understood  about 
the  other  expressions  according  to  the  vowels  there  set. 

32.13  Further  it  is  to  be  known  that  the  first  four  expres- 
sions of  the  first  verse  begin  with  these  consonants,  B,  C,  D, 
F,  and  all  the  subsequent  expressions  begin  with  the  same, 
and  by  this  is  to  be  understood  that  all  the  subsequent  moods 
beginning  with  B  are  reduced  to  the  first  mood  of  the  first 
figure,  with  C  to  the  second,  with  D  to  the  third,  with  F  to 
the  fourth. 

32.14  Further  it  is  to  be  known  that  wherever  S  is  put  in 
these  expressions,  it  signifies  that  the  proposition  understood 
by  means  of  the  vowel  immediately  preceding  should  be 
converted  simply.  And  by  P  is  signified  that  the  proposition 
which  is  understood  by  means  of  the  vowel  immediately 
preceding  is  to  be  converted  accidentally.  And  wherever  M 
is  put,  it  signifies  that  transposition  of  the  premisses  is  to  be 
effected.  Transposition  is  to  make  the  major  minor  and 
conversely.  And  where  C  is  put,  it  signifies  that  the  mood 
understood  by  means  of  that  expression  is  to  be  reduced 
per  impossibile. 

Whence  the  verses: 



S  enjoins  simple  conversion,  per  accidens  P, 
Transpose  with  M,  ad  impossibile  C. 

George  Scholarios's  Greek  version  of  the  four  last  verses  of  32.10 
is  not  without  interest  (cf.  32.02  ff.): 

32.15  Tpa^fjiaTa  Kypa^e  ypacpiSi  re^vixoc;,  (I) 
rpa[X(JLacr!,v  fe'ra^e  x^?lGl  rcapOevos  ispov  (la) 
"Eypoc^s  y.oi'zzjz  uiTpiov  a^oXov.  (II) 
"Ktzolgi  aOevapcx;  taaxt?  a<j7ci§i  6(xaXo<;  (pepwjTos.  (Ill) 

Unlike  the  Latin  ones,  these  verses  are  meaningful,  and  can  be 

Letters  there  wrote  with  a  style  a  scholar, 

With  letters  there  composed  for  the  Graces  a  maiden  a  dedica- 

She  wrote:  Cleave  to  the  moderate,  un-wrathful  (man). 

In  all,  that  strength  which  like  a  shield  is  well-proportioned  is 

the  best. 

The  names  Barbara,  Celareni  etc.  have  survived  the  era  of  Scho- 
lasticism and  are  still  in  use  today,  unlike  many  other  syllogistic 
mnemonics.  We  give  some  examples  of  these  others,  and  first  some 
which  concern  the  technique  of  reducing  moods  of  the  second  and 
third  figures  (also  of  the  'indirect'  moods  of  the  first  figure :  17.111  ff.) 
to  moods  of  the  first.  For  that  purpose,  Jodoc  Trutfeder  at  the 
beginning  of  the  16th  century  gave  these  expressions: 

32.16  Baralipton  Nes-  Celareni 




Dab  His 







































These  expressions  serve  for  the  indirect  process  of  Aristotle 
(§  14,  D).  Thus  for  instance  from  Celareni,  by  putting  the  contra- 
dictory opposite  (/)  of  its  conclusion  (E)  for  the  minor  premiss  (A) 
there  is  concluded  the  contradictory  of  the  latter  (0),  and  one  has 



Festino.  Applying  this  treatment  to  the  major  premiss  one  has 
Disamis.  Evidently  some  further  processes  must  be  employed  to 
get  a  few  further  moods.  E.g.  to  get  Felapton  and  Darapti  from 
Barbara  and  Celarent  respectively,  one  must  first  deduce  Barbari 
and  Celaronl  (24.271  f.).  So  too  in  the  case  of  Dabilis  (24.273). 

Further  mnemonics  that  were  used  will  he  mentioned  later 
(32.24  and  32.38). 


Similar  mnemonic  expressions  are  found  for  the  so-called  'sub- 
alternate'  moods  (24.271-24.281).  A  complete  list  with  names 
appears  in  a  text  of  Peter  of  Mantua  that  is  in  other  respects  very 

32.17  .  .  .  the  first  (formula)  ...  is  usually  signified  by  the 
expression  Barbara.  The  second  formula  has  premisses  arranged 
in  the  way  described  which  conclude  to  the  particular  affir- 
mative or  indefinite  of  the  consequent  of  the  first  formula 
that  we  posited,  and  this  we  are  wont  to  call  Barbari.  .  .  . 

From  the  aforesaid  (premisses)  can  also  be  concluded  the 
particular  negative  of  (i.e.  corresponding  to)  the  consequent 
of  the  aforesaid  formula,  which  (new)  formula  we  can  call 
Celaronl.  .  .  . 

The  eighth  formula,  which  is  called  Baralipton,  follows 
from  Barbari,  by  conversion  of  its  conclusion.  .  .  . 

The  ninth  formula  is  called  Celantes  .  .  .  from  it  follows  the 
tenth,  which  is  called  Celantos;  it  concludes  to  a  particular 
or  indefinite  conclusion.  .  .  . 

The  second  mood  (of  the  second  figure)  can  be  gained  from 
the  aforesaid  premisses  (of  the  mood  Cesare)  by  concluding 
to  the  particular  that  corresponds  to  (Cesare's)  conclusion, 
and  is  signified  by  the  expression  Cesaro.  .  .  . 

The  next  formula  ...  is  usually  called  Cameslres.  From  it 
there  follows  another  formula  which  we  call  Camestro. 

So  Peter  of  Mantua  has  five  subalternate  moods  besides  the 
nineteen  moods  of  Peter  of  Spain,  in  fact  the  full  twenty-four.  But 
he  has  many  others  as  well,  commonly  forming  an  'indirect'  mood 
corresponding  to  each  of  the  others  (applying  the  Aristotelian  rules 
of  §  13),  e.g.  a  Cesares  corresponding  to  Cesare.  Cesares  would  look 
like  this: 

No  man  is  stone ; 
All  marble  is  stone ; 
Therefore  no  man  is  marble. 



Contrast  the  following  in  Camestres : 
All  marble  is  stone; 
No  man  is  stone; 
Therefore  no  man  is  marble. 

The  only  difference  between  the  two  is  in  the  order  of  the  premisses, 
and  to  reckon  them  as  distinct  moods  is  an  extreme  of  formalism. 
Peter  of  Mantua  further  forms  such  moods  as  Barocos,  with  the 
O-conclusion  of  Baroco  converted  (!)  and  other  false  formulas.* 

1.  Among  the  Latins 

We  know  of  no  scholastic  logical  text  in  Latin  where  the  fourth 
figure  in  the  modern  sense  can  be  found,  though  all  logicians  of  the 
period  develop  the  'indirect  moods  of  the  first'.  They  are  mostly 
aware  of  a  fourth  figure,  but  treat  it  as  not  distinct  from  the  first,  e.g. 
Albert  the  Great  (32.18),  Shyreswood  (32.19),  Ockham  (32.20), 
Pseudo-Scotus  (32.21),  Albert  of  Saxony  (32.22),  Paul  of  Venice 
(32.23).  We  give  an  instance  from  Albert  of  Saxony. 

32.22  (The  syllogism  is  constituted)  in  a  fourth  way  if 
the  middle  is  predicated  in  the  first  premiss,  subjected  in  the 
second.  .  .  .  But  it  is  to  be  noted  that  the  first  figure  differs 
from  the  fourth  only  by  interchange  of  premisses  which  does 
nothing  towards  the  deducibility  or  non-deducibility  of  the 

Some  later  logicians  do  recognize  a  'fourth  figure'  but  this  again 
is  not  the  modern  one;  only  the  first  with  interchanged  premisses,  as 
in  the  last  text.  This  is  very  clear  in  Peter  Tarteret  and  Peter  of 
Mantua.  We  quote  the  first: 

32.24  First  (dictum):  Taking  'figure'  in  a  wide  sense,  the 
fourth  figure  is  no  different  from  the  first  but  contained  under 
it.  Second:  taking  'first  figure'  in  a  specific  sense,  a  fourth 
figure  is  to  be  posited  distinct  from  the  first;  and  the  fourth 
figure  consists  in  this,  that  the  middle  is  predicate  in  the 
major  premiss,  subject  in  the  minor,  e.g.  'all  man  is  animal; 
all   animal   is    substance;    therefore   all    man   is   substance'. 

*  It  should  be  understood  in  respect  of  this  and  the  following  sub-section 
that  after  Peter  of  Spain  (generalizing  the  method  of  Boethius  and  Shyreswood 
for  the  second  and  third  figures)  it  was  usual  to  define  the  major  and  minor 
premisses  as  the  first  and  second  stated,  and  the  extreme  terms  with  reference 
to  the  premisses,  not  the  conclusion.  48  moods  in  4  figures  can  be  (and  some- 
times were)  correctly  distinguished  on  this  basis.  'Classical'  failure  to  distinguish 
this  from  the  method  of  Albalag  (which  goes  back  to  Philoponus)  resulted  in 
many  inconsistencies.  (Ed.) 



Third :  there  are  four  moods  of  the  fourth  figure,  viz.  Bamana, 
Camene,  Dimari,  and  Fimeno.  They  are  reduced  to  the  first 
figure  by  mere  exchange  of  premisses. 
Peter  of  Mantua  also  has  Bamana  etc. 

2.  In  Albalag 

Yet  a  clearly  formulated  doctrine  of  the  'genuine'  fourth  figure  is 
to  be  found  in  a  13th  century  text- of  the  Jewish  philosopher  Alba- 
lag.* This  text,  like  the  foregoing,  seems  to  have  been  without 
influence  on  the  development  of  logic  in  the  Middle  Ages.  It  was 
recently  discovered  by  Dr.  G.  Vajda  and  has  never  been  translated 
into  Latin.  We  quote  it  here  at  length  for  its  originality,  and  because 
it  is  instructive  about  the  level  of  logic  at  that  time. 

32.25  In  my  opinion  there  must  be  four  figures.  For  the 
middle  term  can  be  subject  in  one  of  the  two  premisses  and 
predicate  in  the  other  in  two  ways:  (1)  the  middle  term  is 
subject  in  the  minor,  predicate  in  the  major,  (2)  it  is  predicate 
in  the  minor,  subject  in  the  major.  The  ancients  only  con- 
sidered the  second  arrangement  and  called  it  the  'first  figure'. 
This  admits  of  four  moods  which  can  yield  a  conclusion.  But 
the  first  arrangement,  which  I  have  found,  admits  of  five 
moods  which  can  yield  a  conclusion.  .  .  . 

32.26  We  say  then  that  this  new  figure  is  subject  to  three 
conditions:  (1)  one  of  its  premisses  must  be  affirmative,  the 
other  universal;  (2)  if  the  minor  premiss  is  affirmative,  the 
major  will  be  universal;  (3)  if  the  major  is  particular,  the 
minor  will  be  affirmative. 

The  conditions  exclude  eleven  of  the  sixteen  (theoretically 
possible)  moods;  there  remain  therefore  five  which  can  yield 
a  conclusion. 

32.27  (1)  The  minor  particular  affirmative,  the  major 
universal  negative : 

Some  white  is  animal. 

No  raven  is  white. 

Some  animal  is  not  raven. 
Then  one  can  convert  the  minor  particular  affirmative  and 
the  major  universal  negative  and  say: 

Some  animal  is  white, 

No  white  is  raven, 
which  yields  the  third  mood  of  the  first  figure. 

*  This  was  kindly  put  at  my  disposal,  together  with  a  French  translation,  by 
Dr.  G.  Vajda. 



32.28  (2)  The  minor  universal  affirmative,  the  major 
universal  negative: 

All  man  is  animal. 

No  man  is  raven. 

Some  animal  is  not  raven. 
Then  one  can  get  back  to  the  third  mood  of  the  first  figure 
by  converting  both  premisses. 

32.29  (3)  The  minor  universal  negative,  the  major  universal 
affirmative : 

No  man  is  stone. 

Every  speaker  is  man. 

No  stone  is  speaker. 
Exchanging  the  minor  and  major  with  one  another,  one  comes 
back  to  the  second  mood  of  the  first  figure,  of  which  the 
conclusion  will  be:  'No  speaker  is  stone',  and  one  only  needs 
to  convert  this  to  obtain  'No  stone  is  speaker'. 

32.30  (4)  Two  affirmatives: 
All  composite  is  not  eternal. 
All  body  is  composite. 
Some  not  eternal  is  body. 

Here  one  can  interchange  the  minor  and  major  premisses 
and  reach  the  first  figure,  with  conclusion:  'All  body  is  not 
eternal'  which  can  be  converted  to  'Some  not  eternal  is  body'. 

32.31  (5)  The  minor  universal  affirmative,  the  major 
particular  affirmative : 

All  man  is  speaker. 

Some  white  is  man. 

Some  speaker  is  white. 
If   one   interchanges   the   minor   and   major   premisses,    one 
concludes  in  the  first  figure:  'some  white  is  speaker'  which 
will  be  converted  as  above.  .  .  . 

32.32  .  .  .  the  syllogism  is  formed  with  reference  to  a 
determinate  proposition  which  is  first  established  and  laid 
down  in  the  mind,  and  the  truth  of  which  one  then  tries  to 
justify  and  manifest  by  means  of  the  syllogism.  Of  the 
premisses,  that  containing  the  term  which  is  predicate  of 
this  proposition  is  the  major,  that  containing  the  subject 
is  the  minor. 

Albalag  here  presents  the  modern  definition  of  the  syllogistic  terms, 
not  according  to  their  extension,  but  formally,  according  to  their  pla- 
ces in  the  conclusion.  The  modern  names  of  the  moods  he  introduces 
are:  Fresison,  Fesapo,  Calemes,  Bamalip  and  Dimaris  (cf.  §  36,  F). 


assertory  syi.i.oojsth; 

There  is   missing  only  that  corresponding  to   Peter  of  Mantua's 
Celantos  (32.17),  viz.  Calemop.  Albalag  also  formulates  the  general 

rules  of  the  fourth  figure,  and  uses  the  combinatorial  method. 


In  Albert  the  Great  we  find  a  procedure  taken  over  from  the 
Arabs  (32.33)  by  which  all  possible  moods  of  the  syllogism  are  first 
determined  combinatorially,  and  the  invalid  ones  then  discarded. 
The  relevant  text  runs: 

32.34  It  is  to  be  known  that  with  such  an  ordering  of 
terms  and  arrangement  of  premisses  (propositionum)  sixteen 
conjugations  result,  yielded  by  the  quantity  and  quality 
of  the  premisses.  For  if  the  middle  is  subject  in  the  major  and 
predicate  in  the  minor,  either  (1)  both  premisses  are  universal, 
or  (2)  both  are  particular,  or  one  (is)  universal  and  the  other 
particular  and  this  in  two  ways:  for  either  (3)  the  major  is 
universal  and  the  minor  particular,  or  (4)  conversely  the 
major  particular  and  the  minor  universal;  these  are  the  four 
gained  by  combinations  of  quantity.  When  each  is  multiplied 
by  four  in  respect  of  affirmation  and  denial,  there  are  sixteen 
conjugations  in  all,  thus:  if  both  (premisses)  are  universal 
either  (1)  both  are  affirmative,  or  (2)  both  negative,  or  (3) 
the  major  is  affirmative  and  the  minor  negative,  or  con- 
versely (4)  the  major  is  negative  and  the  minor  affirmative: 
and  there  are  four  conjugations.  But  if  both  are  particular, 
there  are  again  four  conjugations  .  .  .  etc. 

We  may  compare  with  that  the  text  of  Albalag  (32.25 ff.).  Kil- 
wardby  uses  similar  methods. 


The  Aristotelian  doctrine  of  the  inveniio  medii  (14.29)  was  keenly 
studied  by  the  Scholastics,  and  the  schema  of  Philoponus  (24.35)  was 
not  only  taken  over,  but  also  further  developed.  It  is  to  be  found 
as  early  as  Albert  the  Great,  who  probably  found  it  in  Averroes 
(32.35) ;  Albert's  version  differs  from  that  of  Philoponus  and  Aver- 
roes only  in  the  particular  formulae  employed.  But  as  we  find  it  in 
him,  it  became  the  foundation  of  the  famous  pons  asinorum.  so 
that  it  must  be  given  in  this  form  as  well : 




All  Pis  M.)  AB 

(NoPisM.)    AD 

(All  Mis  P.)    AC 

FE  (All  S  is  M. 

HE  (NoSisM.) 

(All  MisS.) 

The  further  development  of  this  figure  is  the  pons  asinorum, 
which  must  have  been  known  to  George  of  Brussels  (32.37)  since 
Thomas  Bricot  in  a  commentary  on  George's  lectures  gives  the 
mnemonic  words  for  it  with  the  following  explanation: 

32.38  When  the  letters  A,  E,  I,  0  are  put  in  the  third 
syllable  they  signify  the  quality  and  quantity  of  the  con- 
clusion to  be  drawn.  .  .  .  When  the  letters  A  and  E  are  put 
in  the  first  or  second  syllable,  A  signifies  the  predicate  and  E 
the  subject.  And  each  of  the  letters  can  be  accompanied  by 
three  consonants;  A  with  B,  C,  D,  and  then  B  signifies  that 
the  middle  should  follow  on  the  predicate,  C  that  it  should 
be  antecedent,  D  that  it  should  be  extraneous.  Similarly  E 
is  accompanied  by  F,  G,  H,  and  then  F  signifies  that  the 
middle  should  follow  on  the  subject,  G  that  it  should  be 
antecedent,  H  that  it  should  be  extraneous.  As  is  made 
clear  in  these  verses: 

jB's  the  subject,  F  its  sequent,   G  precedent,  F  outside; 

^4's  the  predicate,  B  its  sequent,  C  precedent,  F  outside. 
.  .  .  To  conclude  to  a  universal  affirmative,  a  middle  is  to 
be  taken  which  is  sequent  to  the  subject  and  antecedent  to 
the  predicate ;  and  this  is  shown  by  Fecana.  ...  To  conclude 
a  particular  affirmative  in  Darapti,  Disamis  and  Dalisi,  a 
middle  is  to  be  taken  which  is  antecedent  to  both  extremes, 
as  is  made  clear  by  Cageti.  ...  To  conclude  to  a  universal 
negative  in  Celarent  or  Cesare  a  middle  is  to  be  taken  which 


frunpergUiitaunipti  y  ■ iituuti 

Pontrm  u-.*Jrjura  (v'rrtrrtuiido  cidci't 
Impcdichicoo.l  •*  fenfm  firm  tc  Jjc  oc  a!tog 
In  doctis  ulcus  cl't  ilhi  nulla  lalus 

Horrrf  rqaus  talcm  ui  ;J.n  i   nc  »itUl?«aC1 
Dun  gradirurccrnrri'  ftc  'kc    ire  potent 
Nod  iguur  rurfum  dira  ueuant  ^finorum 
(^.ui  (cdcos  retro  nunc  rcminrrcuolo. 


/Mmus   .  mratjboin.dump^ns.opanerrrmu       Aims,  Hcti  mrqd  feciSruo  nrcfbftmihiqie^ 
Kficns  5»tiiUiKuadAlab«,uratoicgtcdia*  cadca    Aux.l.uin.mon.corqua.dcdttWFOiOf. 

Pons  Asinorum  after  Peter  Tarteret  (32.39 


is  extraneous  to  the  predicate  and  consequent  on  the  subject, 
as  is  made  clear  by  Dafenes.  But  if  the  inference  is  to  be  in 
Cameslres,  the  middle  must  be  extraneous  to  the  subject  and 
consequent  on  the  predicate,  as  is  made  clear  by  Hebare.  .  .  . 
For  concluding  to  a  particular  negative  in  the  third  figure, 
the  middle  should  be  antecedent  to  the  subject  and  extra- 
neous to  the  predicate,  as  is  made  clear  by  Gedaco.  ...  To 
conclude  indirectly  to  a  particular  affirmative,  the  middle 
must  be  antecedent  to  the  subject  and  consequent  on  the 
predicate,  as  is  made  clear  by  Gebali. 

The  mnemonic  expressions  given  here  employ  the  letters  of 
Albert's  figure,  so  that  each  expression  corresponds  to  a  line  of  the 
figure  having  the  syllables  of  the  mnemonic  at  its  ends,  a  further 
syllable  being  added  to  show  the  quantity  and  quality  of  the 
corresponding  conclusion.  So  this  text  of  George's  both  illustrates 
the  pons  asinorum  and  helps  to  explain  Albert's  figure. 

The  pons  asinorum  itself  we  have  found  only  in  Peter  Tarteret, 
with  the  following  introduction: 

32.39  That  the  art  of  finding  the  middle  may  be  easy, 
clear  and  evident  to  all,  the  following  figure  is  composed 
(ponitur)  to  explain  it.  Because  of  its  apparent  difficulty  it 
is  commonly  called  the  'asses'  bridge'  (pons  asinorum)  though 
it  can  become  familiar  and  clear  to  all  if  what  is  said  in  this 
section  (passu)  is  understood. 

We  give  the  schema  itself  in  the  preceding  illustration. 

The  problem  of  the  null  class,  i.e.  of  the  laws  of  subalternation, 
accidental  conversion  (14.12,  32.08)  and  the  syllogistic  moods 
dependent  on  them,  has  been  much  discussed  in  recent  times 
(46.01  ff.).  It  was  already  posed  in  the  fourteenth  century  and  solved 
by  means  of  the  doctrines  of  supposition  and  appellation.  We  give 
three  series  of  texts,  the  first  attributed  to  St.  Vincent  Ferrer,  the 
second  from  Paul  of  Venice,  the  third  from  a  neo-scholastic  of  the 
17th  century,  a  contemporary  of  Descartes,  John  of  St.  Thomas. 
Each  gives  a  different  solution. 

1.  St.  Vincent  Ferrer 

32.40  Under  every  subject  having  natural  supposition 
copulative  descent  can  be  made,  with  respect  to  the  predicate, 
to  all  its  supposita,  whether  such  a  subject  supposes  discretely* 

*   Reading  discrete  for  difinite. 



or  particularly  or  universally.  Therefore  it  follows  validly: 
Man  is  risible,  therefore  this  man  is  risible,  and  that 
man.  .  .  . 

32.41  But  against  this  rule  there  are  many  objections.  .  .  . 
(Sixth  objection:).  ...  In  the  propositions  'rain  is  water 
falling  in  drops',  'thunder  is  a  noise  in  the  clouds',  the  subjects 
have  natural  supposition.  Yet  it  is  not  always  permissible  to 
descend  in  respect  to  the  predicate  to  the  supposita  of  the 
subject;  for  it  does  not  follow:  'rain  is  water  falling  in  drops, 
therefore  this  rain,  and  that  rain,  etc.';  since  the  antecedent 
is  true  even  when  there  is  no  rain,  as  will  be  shortly  said,  and 
yet  the  consequent  is  not  true  nor  even  very  intelligible, 
since  when  there  is  no  rain  (nulla  pluvia  exislenle)  one  cannot 
say  'this  rain'  or  'that  rain',  since  a  contradiction  would  be 
at  once  implied.  In  the  same  way  must  be  judged  the  proposi- 
tion 'thunder  is  a  noise  etc' 

32.42  To  the  sixth  objection  it  is  to  be  said  that  that  rule  . . . 
is  understood  (to  hold)  when  such  a  subject  has  supposita 
actually  (in  aclu)  and  not  otherwise.  For  no  descent  can  be 
made  to  the  supposita  *  of  anything  except  when  it  has  them 
actually,  since,  as  the  objection  rightly  says,  an  evident  con- 
tradiction would  be  implied.  .  .  .  Hence  the  consequence  which 
concludes  from  a  universal  proposition  to  singular  ones 
contained  under  it,  e.g.  'every  man  runs,  therefore  Socrates 
runs  and  Plato  runs',  and  'every  man  is  an  animal,  therefore 
Plato  is  an  animal  etc.',  is  called  by  some  logicians  'conse- 
quence ul  nunc1.  And  rightly  so,  since  no  such  consequence  is 
valid  except  for  a  determinate  time,  i.e.  when  Socrates  and 
Plato  and  the  other  supposita  actually  exist. 

32.43  Against  the  seventh  objection  it  is  to  be  briefly 
said  that  the  subject  of  the  proposition  'the  rose  is  sweet- 
smelling'  -  or  as  one  can  also  put  it  'the  rose  smells  sweet'  -  has 
personal  supposition,  and  it  follows  validly  'therefore  the  rose 
exists  (est)'.  But  if  one  says  'the  rose  is  odoriferous'  so  that 
'odoriferous'  (hoc  quod  dicitur  odorifera)  expresses  aptitude, 
then  the  subject  has  natural  supposition  and  it  does  not 
follow:  'therefore  the  rose  exists'.  Hence  being  odoriferous  is 
to  the  rose  as  living  to  mankind,  and  what  has  been  said  about 
the  proposition  'man  is  living'  must  also  be  understood  about 
this:  'the  rose  is  odoriferous'. 

*    Reading  supposita  for  subjecla. 



The  solution  of  the  first  text  (32.40-42)  consists  then  precisely  in 
the  exclusion  of  the  null  class  (cf.  §  40,  Bj,  'null'  being  taken  as 
'actually  null'.  In  other  words:  in  the  syllogistic  every  term  must 
have  appellation  in  the  sense  of  Peter  of  Spain  (28.13 f.).   In  the 

second  part  (32.43)  it  is  stipulated  for  subalternation  that  the  subject 
must  have  personal  (27.17 f.)  and  not  natural  (27.14)  supposition. 
This  evidently  presupposes  that  a  term  with  personal  supposition 
stands  for  really  existent  things.  Thus  we  have  the  same  solution 
as  before, 

2.  Paul  of  Venice 

32.44  The  third  rule  is  this:  universal  affirmative  and 
particular  or  indefinite  (affirmative,  as  also  universal  negative 
and  particular  or  indefinite)  negative  (propositions)  which 
have  similarly  and  correctly  supposing  terms,  are  subalternate, 
and  conversely,  explicitly  or  implicitly,  in  the  logical  square 
(figura).  Hence  the  following  are  subalternate:  'every  man  is 
an  animal'  and  'a  certain  man  is  an  animal',  and  similarly: 
'no  man  is  an  animal'  and  'a  certain  man  is  not  an  animal'. 
I  say  'correctly  supposing  terms',  since  the  extremes  must 
explicitly  or  implicitly  stand  for  just  the  same  thing,  if  it  is  a 
case  of  only  one  suppositum,  for  the  same  things,  if  it  is  a 
case  of  several.  And  so  I  say  that  (the  following)  are  not 
subalternate:  'every  man  is  an  animal',  'a  certain  man  is  an 
animal',  since  under  the  supposition  that  there  are  no  men, 
the  universal  would  be  true,  but  the  particular  false,  contrary 
to  the  nature  of  subalterns.  The  reason  why  these  are  not 
subalternate  is  that  the  subjects  do  not  stand  for  exactly 
the  same  thing.  The  subaltern  of  the  former  is  therefore :  'man 
is  an  animal',  and  if  one  required  a  particular  it  must  be  this: 
'a  certain  being  which  is  a  man  is  an  animal'. 

So  Paul  of  Venice  confines  himself  to  stating  the  general  rule 
that  both  propositions  in  a  subalternation  must  have  subjects  with 
exactly  the  same  supposition. 

3.  John  of  St.  Thomas 

John  of  St.  Thomas  deals  with  the  problem  of  conversion. 

32.45  Against  the  conversion  of  the  universal  affirmative, 
it  is  objected:  the  consequence  'every  white  man  is  a  man, 
therefore  a  certain  man  is  a  white  man'  does  not  hold.  For 
the  antecedent  is  necessary,  but  the  consequent  can  be  false, 
in  case  no  man  in  the  world  was  white.  .  .  . 



The  answer  is  that  this  (proposition)  is  not  true  in  the  sense 
in  which  the  first  proposition  of  which  it  is  the  converse  is 
true.  For  when  it  is  said:  'every  white  man  is  a  man',  with 
'is'  taken  accidentally,  for  an  existing  man,  this  proposition 
in  the  argument  given  as  an  example  is  false,  and  its  converse 
too.  But  when  the  'is'  abstracts  from  time  and  renders  the 
proposition  necessary,  then  'white'  will  not  be  verified  in 
the  sense  of  existence,  but  according  to  possibility,  i.e. 
independently  of  time,  in  the  following  sense:  'every  possibly 
white  man  is  a  man',  presupposing  that  no  such  exists. 
Accordingly  the  converse  must  be:  'therefore  a  certain  man 
is  a  man  who  is  possibly  white',  and  thus  this  is  true. 

The  following  may  serve  as  explanation:  take  the  proposition  (1) 
'every  Swiss  king  is  a  man'.  By  the  rules  of  conversion  (14.11, 
32.08)  we  may  infer  (2)  'a  certain  man  is  a  Swiss  king'.  But  (1)  is 
true,  (2)  false.  Therefore  the  rule  of  conversion  employed  is  not 
valid.  To  this  the  Scholastics  would  answer  that  in  (1)  'man'  evi- 
dently stands  for  a  possible  man,  not  for  a  real  one;  it  has  therefore 
no  appellation  in  the  sense  of  28.13.  And  so  if  (1)  is  converted  into 
(2),  'man'  in  (2)  also  supposes  for  possible  men  and  in  this  sense  (2) 
is  true  as  well  as  (1). 

A  further  interesting  point  is  that  singular  terms  always  have 
appellation  (28.13),  so  that  the  Scholastics  attribute  to  proper 
names  the  same  property  with  which  the  moderns  endow  descrip- 
tions (cf.  §  46). 


The  history  of  scholastic  modal  syllogistic  has  been  investigated 
from  the  modern  point  of  view  up  to  and  inclusive  of  Pseudo-Scotus 
(33.01).  We  know  that  there  was  more  than  one  system  of  modal 
logic  in  the  Middle  Ages  and  can  to  some  extent  follow  the  devel- 


The  work  of  Albert  the  Great  constitutes  the  starting  point  and, 
as  his  own  text  suggests  (33.02),  would  seem  to  have  drawn  on 
Arabic  sources.  To  begin  with,  he  shows  much  the  same  doctrine 
as  has  been  ascribed  above  (29.09) ;  cf.  also  23.10,  29.12)  to  Thomas 
Aquinas  and  which  is  basic  for  the  whole  of  Scholasticism  (33.03), 
viz.  the  distinction  of  the  composite  (composita,  de  dido)  and  the 
divided  (divisa,  de  re)  modal  proposition,  i.  e.  between  that  in  which 
the  modal  functor  governs  the  whole  dictum  and  that  in  which  it 



governs  only  a  part.  Later  Albert  gives  a  clear  statement  of  the 
Aristotelian  distinction  of  the  two  structure-  of  the  modal  proposi- 
tion in  the  divided  sense: 

33.04  That  the  predicate  A  possibly  belongs  to  the  subject 
B  means  one  of  these:  (1)  that  A  possibly  belongs  to  that 
which  is  B  and  of  which  B  is  predicated  in  the  sense  of  actual 
inherence,  or  (2)  that  A  possibly  belongs  to  that  to  which  B 
possibly  belongs. 

There  is  added  a  third  structure,  unknown  to  Aristotle: 

33.05  And  if  someone  asks  why  the  third  meaning  (acceptio) 
of  the  contingent  is  not  given  here,  viz.  that  whatever  is 
necessarily  B  is  possibly  A,  since  this  is  used  in  the  mixing  of 
the  contingent  and  the  necessary,  it  is  to  be  answered  that  it 
is  left  sufficiently  clear  from  what  else  has  been  determined 
about  the  mixing  of  the  assertoric  and  the  contingent. 

The  structure  in  question  is  this: 

For  all  x:  if  x  is  necessarily  B,  x  is  possibly  A. 

It  is  significant  that  Albert  the  Great  puts  this  doctrine  at  the 
beginning  of  the  presentation  of  his  theory  of  modal  syllogisms  in  a 
special  chapter  entitled  De  dici  de  omni  el  dici  de  nullo  in  propo- 
silionibus  de  contingenti  (33.06).  What  for  Aristotle  are  marginal 
thoughts  about  the  structure  of  modals  by  comparison  with  his 
main  ideas  (15.13),  have  here  become  fundamental. 

We  find  then  in  Albert  the  Great  a  systematization  of  the  Ari- 
stotelian teaching  about  the  kinds  of  modal  functors  (33.07),  and  a 
thorough  presentation  of  the  syllogistic  of  the  Prior  Analytics. 


Besides  the  four  classical  modal  functors  Pseudo-Scotus  intro- 
duces others:  'of  itself  (per  se),  'true',  'false',  'doubtful'  (dubium), 
'known'  {scitum),  'opined'  (opinatum),  'apparent',  'known'  (notum), 
'willed'  (volitum),  'preferred'  (dilectum)  (33.08),  and  so  a  number  of 
'subjective'  functors.  He  formulates  a  long  series  of  laws  of  modal 
propositional  logic  (modal  consequences),  among  which  are  the 

33.09  If  the  antecedent  is  necessary  the  consequent  is 
necessary  .  .  .  and  similarly  with  the  other  (positive)  modes. 

33.10  Modal  (de  modo)  propositions  in  the  composite  sense 
with  the  (negative)  modes  'impossible',  'false',  'doubtful'  are 
not  convertible  like  assertoric  ones.  Proof:  for  otherwise  the 



(following)  rules  would  be  true :  'if  the  antecedent  is  impossible, 
the  consequent  is  impossible',  'if  the  antecedent  is  doubtful, 
the  consequent  is  doubtful' ;  but  they  are  false.  .  .  . 

33.11  It  follows:  possibly  no  B  is  A,  therefore  possibly  no 
A  is  B:  since  both  (propositions)  'no  B  is  A'  and  'no  A  is  B' 
follow  from  one  another.  So  if  one  is  contingent,  the  other  is 
too:  otherwise  the  contingent  would  follow  from  the  neces- 
sary. .  .  . 

33.12  If  the  premisses  are  necessary,  the  conclusion  is 

With  the  help  of  these  and  other  laws  known  to  us  from  the 
chapter  on  propositional  consequences,  Pseudo-Scotus  proceeds 
to  establish  two  systems  of  syllogistic,  one  with  modal  propositions 
in  the  composite,  the  other  in  the  divided  sense.  As  premisses  he 
uses  not  only  contingent  but  also  (one-sidedly:  (15.071)  possible 
and  impossible  propositions.  We  cite  only  a  few  examples  from  his 
teaching  on  conversion: 

33.13  Modal  proposition  in  the  composite  sense  are  con- 
verted in  just  the  same  way  as  assertoric  ones. 

33.14  Affirmative  possible  (de  possibili)  propositions  in  the 
divided  sense  (in  which  the  subject  stands)  for  that  which  is, 
are  not  properly  speaking  converted.  Proof:  on  the  supposition 
that  whatever  is  in  fact  running  is  an  ass,  the  following  is 
true :  'every  man  can  run'  in  the  sense  that  everything  which 
is  a  man  is  able  to  run,  but  its  converse  is  false:  'a  certain 
runner  can  be  a  man',  .  .  .  And  I  say  'properly  speaking'  on 
purpose,  since  (these  propositions)  can  in  a  secondary  sense 
be  converted  into  assertorics.  E.g.  'every  man  can  run, 
therefore  a  certain  thing  that  can  run  is  a  man'.  .  .  . 

33.15  The  third  thesis  concerns  affirmative  possible  pro- 
positions, in  which  the  subject  stands  for  that  which  can 
exist,  for  such  affirmatives  are  converted  in  the  same  way  as 
assertorics.  .  .  . 

33.16  As  concerns  necessary  propositions,  and  first  those 
which  are  to  be  understood  in  this  (divided)  sense  with  subject 
standing  for  what  is,  the  first  thesis  is  this,  that  affirmatives.  .  . 
are  not  converted;  for  supposing  that  God  is  creating,  it 
does  not  follow:  'whatever  is  creating  is  necessarily  God, 
therefore  a  certain  God  is  creating  necessarily'. 

33.17  The  second  thesis  is  that  negative  necessary  propo- 
sitions with  subject  standing  for  what  is,  are  not  converted.  . .  . 



33.18  Third  thesis:  that  affirmative  necessary  propositions 
with  subject  standing  for  what  can  be  are  not  properly 
speaking  converted.  .  .  . 

33.19  Fourth  thesis  about  necessary  propositions  with 
subject  standing  for  what  can  be :  universal  negatives  are 
simply  converted,  particular  negatives  not.  Proof:  since,  as 
has  been  said  earlier,  the  particular  affirmative  possible 
(proposition  with  subject  standing)  for  what  can  be,  converts 
simply,  and  it  contradicts  the  universal  negative  necessary 
(proposition  with  subject  standing)  for  what  can  be;  so:  if 
one  of  two  contradictories  is  simply  converted,  so  is  the  other, 
since  when  the  consequent  follows  from  the  antecedent,  the 
opposite  of  the  antecedent  follows  from  the  opposite  of  the 


Pseudo-Scotus  introduced  1.  one-sidedly  possible  premisses  into 
the  syllogistic,  2.  in  the  composite  sense.  Ockham  has  a  further 
innovation:  he  treats  also  of  syllogisms  in  which  one  premiss  is 
taken  in  the  composite,  the  other  in  the  divided  sense.  At  the  same 
time  the  whole  modal  syllogistic  is  formally  developed  from  its 
structural  bases  with  remarkable  acumen.  We  give  only  two 

33.20  As  to  the  first  figure  it  is  to  be  known  that  when 
necessary  premisses  are  taken  in  the  composite  sense,  or 
when  some  are  taken  that  are  equivalent  to  those  propositions 
in  the  composite  sense,  there  is  always  a  valid  syllogism 
with  a  conclusion  that  is  similar  in  respect  of  the  composite  or 
equivalent  sense.  .  .  .  But  when  all  the  propositions  are  taken 
in  the  divided  sense,  or  equivalent  ones,  a  direct  conclusion 
always  follows,  but  not  always  an  indirect.  The  first  is  evident 
because  every  such  syllogism  is  regulated  by  did  de  omni  or 
[did)  de  nullo.  For  by  such  a  universal  proposition  it  is  denoted 
that  of  whatever  the  subject  is  said,  of  that  the  predicate  is 
said.  As  by  this:  'every  man  is  necessarily  an  animal'  is 
denoted  that  of  whatever  the  subject  'man'  is  said,  of  that  the 
predicate  'animal'  is  necessarily  said.  And  the  same  holds 
good  proportionately  of  the  universal  negative.  Therefore 
adjoining  a  minor  affirmative  in  which  the  subject  (of  the 
major)  is  predicated  of  something  with  the  mode  of  necessity, 
the  inference  proceeds  by  did  de  omni  or  de  nullo  (cf.  14.23). 
Hence  it  follows  validly:  'every  man  is  necessarily  an  animal, 



Socrates  is  necessarily  a  man,  therefore  Socrates  is  necessarily 
an  animal'.  But  the  indirect  conclusion,  viz.  the  converse  of 
that  conclusion  with  no  other  variation  than  the  transposition 
of  terms  does  not  follow  .  .  .  (There  follows  here  the  reason  as 
in  33.16).  But  if  the  major  is  taken  in  the  composite  sense, 
or  an  equivalent  (proposition),  and  the  minor  in  the  divided 
sense,  the  conclusion  follows  in  the  composite  sense  and  not  in 
the  divided.  The  first  is  evident,  because  it  follows  validly: 
'This  is  necessary:  every  divine  person  is  God,  one  creating  is 
necessarily  a  divine  person,  therefore  one  creating  is  necessa- 
rily God'.  But  this  does  not  follow:  'therefore  this  is  necessary: 
one  creating  is  God'. 

But  if  the  major  is  taken  in  the  divided  sense,  or  an  equi- 
valent (proposition),  and  the  minor  in  the  composite  sense, 
the  conclusion  follows  in  the  divided  sense  and  in  the  composite 
sense.  And  the  reason  is  that  it  is  impossible  that  something 
(B)  essentially  (per  se)  or  accidentally  inferior  (to  A)  should 
be  necessarily  predicated  of  something  (67),  so  that  the 
proposition  ('67  is  B')  would  be  necessary,  without  the 
proposition  in  which  the  superior  (A)  of  that  inferior  (B)  is 
predicated  of  the  same  (67)  (viz.  '67  is  A')  also  being  neces- 

So  for  every  Aristotelian  formula  Ockham  has  the  four:  (1)  with 
both  premisses  in  the  composite  sense,  like  Theophrastus  (cf. 
17.15  f.);  (2)  with  both  premisses  in  the  divided  sense,  like 
Aristotle  -  as  was  concluded  from  the  indications  available  (§  15,  B) ; 
(3)  the  major  premiss  composite,  the  minor  divided ;  (4)  the  major 
premiss  divided,  the  minor  composite. 

Another  example  is  the  following  treatment  of  syllogisms  with 
both  premisses  in  the  mode  of  simple  (one-sided)  possibility: 

33.21  ...  I  here  understand  'possible'  in  the  sense  of  the 
possibility  which  is  common  to  all  propositions  that  are  not 
impossible.  And  it  is  to  be  known  that  in  every  figure,  if  all 
the  propositions  be  taken  as  possible  in  the  compounded 
sense,  or  if  equivalent  ones  to  those  be  taken,  the  syllogism  is 
invalid  because  inference  would  proceed  by  this  rule:  the 
premisses  are  possible,  therefore  the  conclusion  is  possible, 
which  rule  is  false.  Hence  it  does  not  follow:  'that  everything 
coloured  is  white  is  possible,  that  everything  black  is  coloured 
is  possible,  therefore  that  everything  black  is  white  is  pos- 
sible'. .  .  .  And  so  the  rule  is  false:  the  premisses  are  possible 



therefore  the  conclusion  is  possible.  But  this  rule  is  true:  if 
the  premisses  are  possible  and  compossible,  the  conclusion  is 
possible  (cf.  31.27)  .  .  .  But  if  the  possible  proposition  be  taken 
in  the  divided  sense,  or  an  equivalent  one  be  taken,  such  as  are 
propositions  like  'every  man  can  be  white',  'a  white  thing  can 
be  black'  etc.  .  .  .  there  the  subject  can  stand  for  things 
which  are  or  for  things  which  can  be,  i.e.  for  things  of  which  it 
is  verified  by  a  verb  in  the  present,  or  for  things  of  which  it  is 
verified  by  a  verb  of  possibility.  ...  As  if  I  say:  'every  white 
thing  can  be  a  man',  one  sense  is  this:  everything  which  is 
white  can  be  a  man,  and  this  sense  is  true  if  there  be  nothing 
white  but  man.  Another  sense  is  this:  everything  which  can 
be  white  can  be  a  man,  and  this  is  false  whether  only  man  be 
white  or  something  other  than  man.  .  .  . 

And  it  is  to  be  known  that  if  the  subject  of  the  major  be 
taken  for  things  which  can  be  .  .  .  however  the  subject  of  the 
minor  be  taken,  the  uniform  syllogism  is  always  valid  and  is 
regulated  by  did  de  omni  or  de  nullo,  and  the  common  prin- 
ciples of  the  assertoric  syllogism  hold.  E.g.  if  one  argues  thus: 
'every  white  thing  can  be  a  man  -  i.e.  everything  that  can  be 
white  can  be  a  man  -  every  ass  can  be  white,  therefore  every 
ass  can  be  a  man'.  .  .  . 

But  if  the  subject  of  the  major  supposes  for  things  which 
are,  then  such  a  uniform  (syllogism)  is  not  valid,  for  it  does  not 
follow:  'everything  which  is  white  can  be  a  man,  every  ass 
can  be  white,  therefore  every  ass  can  be  a  man'.  For  if  there 
be  nothing  white  but  man,  the  premisses  are  true  and  the 
conclusion  false.  .  .  . 

These  examples  may  suffice  to  give  an  idea  of  the  problems 

We  now  give  a  summary  of  the  different  kinds  of  modal  syllogism 
which  Ockham  considered.  He  distinguishes  the  following  functors 
and  kinds  of  functor:  (1)  'necessary',  (2)  'possible'  (one-sidedly), 
(3)  'contingent'  (two-sidedly,  (4)  'impossible',  (5)  other  modes 
(subjective).  Further  there  are  (6)  the  assertoric  propositions.  Ock- 
ham deals  with  syllogisms  with  premisses  in  the  following  combi- 





















Altogether  then  he  has  eighteen  classes.  In  each  he  discusses  the 
four  formulae  mentioned  above,  and  this  in  each  of  three  figures  -  the 
analogates  therefore  of  the  nineteen  classical  moods.  Theoretically 
this  gives  1368  formulae,  but  many  of  them  are  invalid. 

Here,  however,  as  with  Aristotle  (§  15,  D),  there  are  also  many 
moods  without  analogues  in  the  assertoric  syllogistic,  so  that  the 
total  number  of  valid  modal  syllogisms  for  Ockham,  in  spite  of  the 
many  invalid  analogues,  may  reach  about  a  thousand. 


The  Scholastics  did  not  look  on  propositions  about  the  future 
and  the  past  as  modal,  but  they  treated  them  quite  analogously  to 
modals.  Two  texts  from  Ockham  illustrate  this  point: 

33.22  Concerning  the  conversion  of  propositions  about  the 
past  and  the  future,  the  first  thing  to  be  known  is  that  every 
proposition  about  the  past  and  the  future,  in  which  a  common 
term  is  subject,  is  to  be  distinguished  ...  in  that  the  subject 
can  suppose  for  what  is  or  for  what  has  been,  if  it  is  a  proposi- 
tion about  the  past  .  .  .  e.g.  'the  white  thing  was  Socrates'  is 
to  be  distinguished,  since  'white'  can  suppose  for  what  is 
white  or  for  what  was  white.  But  if  the  proposition  is  about 
the  future,  it  is  to  be  distinguished  because  the  subject  can 
suppose  for  what  is  or  for  what  will  be.  .  .  .  Secondly  it  is  to  be 
known  that  when  the  subject  of  such  a  proposition  supposes  for 
what  is,  then  the  proposition  should  be  converted  into  a 
proposition  about  the  present,  the  subject  being  taken  with 
the  verb  'was'  and  the  pronoun  'which',  and  not  into  a 
proposition  about  the  past.  Hence  this  consequence  is  not 
valid :  'no  white  thing  was  a  man,  therefore  no  man  was  white', 
if  the  subject  of  the  antecedent  be  taken  for  what  is.  For  let 
it  be  supposed  that  many  men  both  living  and  dead  have  been 
white,  and  that  many  other  things  are  and  have  been  white, 
and  that  no  man  is  now  white,  then  the  antecedent  is  true  and 
the  consequent  false.  .  .  .  And  so  it  should  not  be  converted  as 
aforesaid  but  thus:  'no  white  thing  was  a  man,  therefore 
nothing  which  was  a  man  was  white'. 

Then  an  example  from  syllogistic : 

33.23  Now  we  must  see  how  syllogisms  are  to  be  made 
from  propositions  about  the  past  and  the  future.  Here  it  is 
to  be  known  that  when  the  middle  term  is  a  common  term,  if 
the  subject  of  the  major  supposes  for  things  which  are,  the 



minor  should  be  about  the  present  and  not  the  future  or  the 
past;  for  if  the  minor  proposition  was  about  the  past  and  not 
the  present  such  a  syllogism  would  not  be  governed  by  did  de 
omni  or  de  nullo,  because  in  a  universal  major  about  the  past 
with  subject  supposing  for  things  which  are,  it  is  not  denoted  that 
the  predicate  is  affirmed  or  denied  by  the  verb  in  the  past  about 
whatever  the  subject  is  affirmed  of  by  the  verb  in  the  past.  But 
it  is  denoted  that  the  predicate  is  affirmed  or  denied  by  the  verb 
in  the  past  about  whatever  the  subject  is  affirmed  of  by  a  verb 
in  the  present.  .  .  .  But  if  the  subject  of  the  major  supposes  for 
things  which  have  been,  then  one  should  not  adjoin  a  minor 
about  the  present,  because  as  is  quite  evident,  the  inference 
does  not  proceed  by  did  de  omni  or  de  nullo;  but  a  minor  about 
the  past  should  be  taken,  and  it  makes  no  difference  whether 
the  subject  of  the  minor  supposes  for  things  which  are  or 
things  which  have  been.  Hence  this  syllogism  is  invalid: 
'every  white  thing  was  a  man,  an  ass  is  white,  therefore  an 
ass  was  a  man'.  .  .  .  What  has  been  said  about  propositions 
concerning  the  past,  is  to  be  maintained  proportionately  for 
those  about  the  future. 

These  principles  are  then  applied  to  the  syllogisms  in  the  different 


In  view  of  what  we  know  about  e.g.  the  composite  and  divided 
senses  (29.13),  and  of  our  occasional  discoveries  of  similar  doctrines 
(28.15ff.),  we  must  suppose  that  the  Scholastics  developed  a  num- 
ber of  logical  theories  not  pertaining  either  to  propositional  logic  or  to 
syllogistic  in  the  Aristotelian  sense.  But  this  field  is  hardly  at  all 
explored ;  e.g.  we  do  not  know  whether  they  were  acquainted  with  a 
more  comprehensive  logic  of  relations  than  that  of  Aristotle. 

We  cite  a  few  texts  belonging  to  such  theories,  viz.  (1)  a  series  of 
texts  about  non-Aristotelian  'syllogisms'  with  singular  terms,  (2)  an 
analysis  of  the  quantifiers  'ever'  and  'some',  (3)  a  'logical  square' 
of  so-called  'exponible'  propositions,  i.e.  of  propositions  equivalent 
to  the  product  or  sum  of  a  number  of  categoricals.  In  that  connec- 
tion we  finally  give  some  theorems  about  the  so-called  syllogismus 
obliquus,  which  was  not  without  importance  for  the  later  history  of 

Here  it  must  be  stressed  even  more  than  usual,  that  these  are 
only  fragments  concerning  a  wide  range  of  problems  that  has  not 
been  investigated. 




A  first  widening  of  the  Aristotelian  syllogistic  consists  in  the 
admission  of  singular  terms  and  premisses.*  Ockham  already  knows 
of  the  substitution  that  was  to  become  classic : 

34.01  Every  man  is  an  animal; 
Socrates  is  a  man; 

Therefore,  Socrates  is  an  animal. 

Here  the  minor  premiss  is  singular.  But  Ockham  also  allows 
singular  propositions  as  major  premisses: 

34.02  For  it  follows  validly  (bene):  'Socrates  is  white, 
every  man  is  Socrates,  therefore  every  man  is  white'.  .  .  .  And 
such  a  syllogism  ...  is  valid,  like  that  which  is  regulated  by 
did  de  omni  or  de  nullo,  since  just  as  the  subject  of  a  universal 
proposition  actually  stands  for  all  its  significates,  so  too  the 
singular  subject  stands  for  all  its  significates,  since  it  only 
has  one. 

The  difference  between  a  syllogism  as  instanced  in  34.01  and  the 
classical  Aristotelian  syllogism  is  only  'purely  verbal'  (34.03)! 

This  may  well  be  termed  a  revolutionary  innovation.  Not  only  are 
singular  terms  admitted,  contrary  to  the  practice  of  Aristotle,  but 
they  are  formally  equated  with  universal  ones.  The  ground  advanced 
for  this  remarkable  position  is  that  singular  terms  are  names  of 
classes,  just  like  universal  terms,  only  in  this  case*  unit-classes.  There- 
fore 34.02  is  not  propounding  the  syllogism  as  a  substitution  in  the 

34.021  If  'for  all  x:  if  x  is  an  S  then  x  is  a  P'  holds,  and 
la  is  an  S'  holds,  then  it  also  holds :  'a  is  a  P'. 

-  where  'S'  and  'P'  are  to  be  thought  of  as  class-names,  'a'  as  an 
individual  name  -,  but  as  a  substitution  in: 

34.022  If  'for  alia;:  if  x  is  an  M  then  x  is  a  P'  holds,  and  'for 
all  x:  if  x  is  an  5  then  x  is  an  M'  holds,  then  it  holds:  'for  all 
x:  if  x  is  an  S  then  x  is  a  P' 

-  where  'M',  'S'  and  'P'  are  all  class-names.  In  that  case  the  sole 
difference  between  the  Aristotelian  and  Ockhamist  syllogisms  is  that 
the  former  is  a  proposition,  the  latter  a  rule.  Admittedly  the  basis  of 
the  system  is  altered  with  the  introduction  of  names  for  unit- 

Again,  the  syllogisms  with  singular  terms  that  are  usually  attri- 
buted to  Peter  Ramus,  are  already  to  be  found  in  Ockham. 
34.02  contains  one  example;  here  is  another: 

*   But  34.01  is  Stoic.  Vid.  Sextus  Empiricus,  Pyrr.  Hyp.  B  164  ff. 



34.04  Although  it  has  been  said  above  that  one  cannot  argue 
from  affirmatives  in  the  second  figure,  yet  two  cases  are  to  be 
excepted  from  that  general  rule.  The  first  is,  if  the  middle  term 
is  a  discrete  term,  for  then  one  can  infer  a  conclusion  from 
two  affirmatives,  e.g.  it  follows  validly:  'every  man  is  Socrates, 
Plato  is  Socrates,  therefore  Plato  is  a  man'.  And  such  a 
syllogism  can  be  proved,  because  if  the  propositions  are  con- 
verted there  will  result  an  expository  syllogism  in  the  third 

The  proof  offered  at  the  end  of  that  text  is  evidently  connected 
with  the  Aristotelian  ecthesis  (13.13),  as  is  suggested  by  the  scholastic 
term  'expository  syllogism'  and  the  following  text  from  Ockham: 

34.05  Besides  the  aforesaid  syllogisms,  there  are  also 
expository  syllogisms,  about  which  we  must  now  speak. 
Where  it  is  to  be  known  that  an  expository  syllogism  is  one 
which  is  constituted  by  two  singular  premisses  arranged  in 
the  third  figure,  which,  however,  can  yield  both  a  singular, 
and  a  particular  or  indefinite  conclusion,  but  not  a  universal 
one,  just  as  two  universals  in  the  third  figure  cannot  yield  a 
universal.  ...  To  which  it  must  be  added  that  the  minor 
must  be  affirmative,  because  if  the  minor  is  negative  the 
syllogism  is  not  valid.  ...  If  the  minor  is  affirmative,  whether 
the  major  is  affirmative  or  negative,  the  syllogism  is  always 

Stephen  de  Monte  summarizes  this  doctrine  in  systematic 
fashion : 

34.06  But  it  is  asked  whether  we  can  rightly  syllogize  by 
means  of  an  expository  syllogism  in  every  figure ;  I  say  that  we 
can.  For  affirmatives  hold  in  virtue  of  this  principle :  when  two 
different  terms  are  united  with  some  singular  term  taken 
singularly  and  univocally,  in  some  affirmative  copulative 
proposition  from  which  the  consequence  holds  to  two  uni- 
versal affirmatives  (de  omni),  such  terms  should  be  mutually 
united  in  the  conclusion.  .  .  .  But  negatives  hold  in  virtue  of 
this  principle:  whenever  one  of  two  terms  is  united  with  a 
singular  term  etc.,  truly  and  affirmatively,  and  the  other 
negatively,  such  terms  should  be  mutually  united  negatively, 
respect  being  had  to  the  logical  properties.  .  .  . 

Seven  syllogisms  arise  in  this  way,  two  in  each  of  the  first  and 
third  figures,  three  in  the  second. 


34.07  We  proceed  to  the  signs  which  render  (propositions) 
universal  or  particular.  ...  Of  such  signs,  one  is  the  universal, 
the  other  is  the  particular.  The  universal  sign  is  that  by  which 
it  is  signified  that  the  universal  term  to  which  it  is  adjoined 
stands  copulatively  for  its  suppositum  (per  modum  copula- 
tionis).  .  .  .  The  particular  sign  is  that  by  which  it  is  signified 
that  a  universal  term  stands  disjunctively  for  all  its  supposita. 
And  I  purposely  say  'copulatively'  when  speaking  of  the  uni- 
versal sign,  since  if  one  says :  'every  man  runs  'it  follows  formal- 
ly: 'therefore  this  man  runs,  and  that  man  runs,  etc'  But  of 
the  particular  sign  I  have  said  that  it  signifies  that  a  universal 
term  to  which  it  is  adjoined  stands  disjunctively  for  all  its 
supposita.  That  is  evident  since  if  one  says:  'some  man  runs' 
it  follows  that  Socrates  or  Plato  runs,  or  Cicero  runs,  and  so 
of  each  (de  singulis).  This  would  not  be  so  if  this  term  did 
not  stand  for  all  these  (supposita);  but  it  is  true  that  this  is 
disjunctive.  Hence  it  is  requisite  and  necessary  for  the  truth 
of  this:  'some  man  runs',  that  it  be  true  of  some  (definite) 
man  to  say  that  he  runs,  i.e.  that  one  of  the  singular  (pro- 
positions) is  true  which  is  a  part  of  the  disjunctive  (proposi- 
tion) :  'Socrates  (runs)  or  Plato  runs,  and  so  of  each',  since  it  is 
sufficient  for  the  truth  of  a  disjunctive  that  one  of  its  parts  be 
true  (cf.  31.10  and  31.223). 

This  is  the  quite  'modern'  analysis  of  quantified  propositions 
(44.03)  in  the  following  equivalences: 

34.071  (For  all  x:  x  is  F)  if  and  only  if:  [a  is  F)  and  (b  is  F) 
and  (c  is  F)  etc. 

34.072  (There  is  an  x  such  that  x  is  F)  if  and  only  if: 
{a  is  F)  or  (b  is  F)  or  (c  is  F)  etc. 

Further  remarkable  is  the  express  appeal  to  a  propositional  rule. 
In  this  text  propositional  logic  is  consciously  made  the  basis  of  term- 
logic,  and  this  is  only  one  of  many  examples. 

The  so-called'exponible'  propositions  were  scholastically  discussed 
in  considerable  detail.  They  are  those  which  are  equivalent  to  a 
product  or  sum  of  a  number  of  categoricals.  There  are  three  kinds, 
the  exclusive,  the  exceptive,  and  the  reduplicative.  In  view  of  the 
metalogical  treatment  we  give  the  'logical  squares'  of  Tarteret  for  the 
first  two  kinds,  with  a  substitution,  also  from  him,  and  the  mnemonic 



34.08  DIVES 

'Only  man  is  an 

animal'  is  thus 

expounded :        CONTRARY 

(1)  man  is  an 

animal  and  (2) 

nothing  which  is 

not  man  is  an 



'Only  man  is 
not  an  animal' : 

(1)  man  is  not 
an  animal  and 

(2)  everything 
that  is  not  man 
is  an  animal. 




















'Not  only  man  is 
not  an  animal' : 
(1)  every  man  is 
an  animal  or  (2) 
something  which 
is  not  a  man  is 
not  an  animal 



'Not  only  man 
is  an  animal' : 

(1)  man  is  not 
an  animal  or 

(2)  something 
which  is  not 
man  is  an  animal. 


34.09  AM  ATE 

'Every  man  besides 
Socrates  runs' : 


'Every  man  besides 
Socrates  does  not 
(1)  every  man  who  (1)  every  man  who 

is  not  Socrates         CONTRARY  is  not  Socrates  does 
runs  and  (2)  not  run  and  (2) 

Socrates  is  a  man  Socrates  is  a  man 

and  (3)  Socrates  and  (3)  Socrates 

does  not  run.  runs. 



O                                                      4y 

'Not  every  man             SUB-          'Not  every  man 
besides  Socrates    CONTRARY     besides  Socrates 

does  not  run' : 

runs' :  (1)  some 

(1)  some  man  who 

man  who  is  not 

is  not  Socrates 

Socrates  does  not 

runs,  or  (2)  Socra- 

run or  (2)  Socrates 

tes  is  not  a  man  or 

is  not  a  man  or 

(3)  Socrates  does 

Socrates  runs. 

not  run. 



The  originality  of  the  formal  laws  by  substitution  in  which  the 
consequences  shown  in  these  squares  are  gained,  consists  in  their 
being  a  combination  of  the  theory  of  consequences  (especially  the 
so-called  'de  Morgan'  laws,  cf.  31.35 f.)  with  the  Aristotelian  doctrine 
of  opposition  (logical  square:  12.09 f.).  They  are  all  valid,  and  one 
can  only  marvel  at  the  acumen  of  those  logicians  who  knew  how 
to  deduce  them  without  the  aid  of  a  formalized  theory.  How  com- 
plicated are  the  processes  of  thought  underlying  the  given  schemata 
can  be  shown  by  one  of  the  simplest  examples,  in  which  ANNO 
follows  from  D IVES.  DIVES  must  be  interpreted  thus : 

(1)  Some  M  is  L,  and:  no  not-M  is  L.  From  that  there  follows  by 
the  rule  31.222: 

(2)  no  not-M  is  L, 

and  from  that  in  turn,  by  the  law  of  subalternation  (24.29,  cf.  32.44) : 

(3)  some  not-M  is  not  L. 

Applying  the  rule  31.10  (cf.  31.223)  one  obtains  : 

(4)  every  M  is  L  or  some  not-M  is  not  L  which  was  to  be  proved. 


The  Aristotelian  moods  with  'indirect'  premisses  (16.24ff.)  were 
also  systematically  elaborated  and  developed  by  the  Scholastics. 
Ockham  (34.10)  already  knew  more  than  a  dozen  formulae  of  this 
kind.  But  so  far  as  we  know,  no  essentially  new  range  of  problems 



was  opened  up.  We  cite  some  substitutions  in  such  moods  from 
Ockham;  their  discovery  has  been  quite  groundlessly  attributed  to 

34.11  It  also   follows  validly:   'every  man  is  an  animal, 
Socrates  sees  a  man,  therefore  Socrates  sees  an  animal'. 

34.12  It  follows  validly:  'every  man  is  an  animal,  an  ass 
sees  a  man,  therefore  an  ass  sees  an  animal'. 

34.13  It  follows:  'no  ass  belongs  to  man,  every  ass  is  an 
animal,  therefore  some  animal  does  not  belong  to  man'. 



Concerning  also  the  search  for  solutions  of  antinomies  in  the  Middle 
Ages  insufficient  knowledge  is  available  for  us  to  be  able  to  survey 
the  whole  development  here,  though  J.  Salamucha  devoted  a  serious 
paper  to  it  (35.01).  The  connected  problems  seem  to  have  been  well 
known  in  the  middle  of  the  13th  century,  but  without  their  impor- 
tance being  realized.  Albert  the  Great  merely  repeats  the  Aristotelian 
solution  of  the  Liar  (35.02),  and  again  Giles  of  Rome  (in  the  second 
half  of  the  13th  century)  only  treats  this  antinomy  briefly  and  quite 
in  the  Aristotelian  way  (35.03).  Peter  of  Spain,  whose  Summulae 
treat  of  all  the  problems  then  considered  important,  considers  the 
fallacy  of  what  is  'simply  and  in  a  certain  respect'  (under  which 
heading  Aristotle  deals  with  the  Liar,  cf.  23.18)  (35.04),  but  says 
nothing  about  antinomies. 

However,  two  points  are  worth  noting  about  Albert  the  Great;  he 
is  the  first  that  we  find  using  the  expression  'insoluble'  (insolubile) 
which  later  became  a  technical  term  in  this  matter,  and  then  he  has 
some  formulations  that  are  new,  at  least  in  detail.  This  can  be  seen 
in  a  passage  from  his  Elenchics : 

35.05  I  call  'insoluble'  those  (propositions)  which  are  so 
formed  that  whichever  side  of  the  contradictory  is  granted, 
the  opposite  follows.  .  .  e.g.  someone  swears  that  he  swears 
falsely;  he  swears  either  what  is  true,  or  not.  If  he  swears  that 
he  swears  falsely,  and  swears  what  is  true,  viz.  that  he  swears 
falsely,  nobody  swears  falsely  in  swearing  what  is  true :  there- 
fore he  does  not  swear  falsely,  but  it  was  granted  that  he 
does  swear  falsely.  But  if  he  does  not  swear  falsely  and  swears 
that  he  swears  falsely,  he  does  not  swear  what  is  true ;  there- 



fore  he  swears  falsely:  because  otherwise  he  would  not  swear 
what  is  true  when  he  swears  that  he  swears  falsely. 

By  the  time  of  Pseudo-Scotus  the  subject  has  become  a  burning 
one;  he  cites  at  least  one  solution  that  diverges  from  his  own  (35.06) 
and  treats  the  question  in  two  chapters  of  which  the  first  bears  the 
title  'Whether  a  universal  term  can  stand  for  the  whole  proposition 
of  which  it  is  a  part'  (35.07).  The  answer  is  a  decisive  negative : 

35.08  It  is  to  be  said  that  a  part  as  part  cannot  stand  for 
the  whole  proposition. 

His  solution,  however,  does  not  consist  in  an  application  of  this 
principle,  but  is  found  in  the  distinction  between  the  signified  and  the 
exercised  act: 

35.09  If  it  is  said:  'I  say  what  is  false,  therefore  it  is  true 
that  I  say  what  is  false',  I  answer  that  the  consequence  does 
not  hold  formally,  as  (also)  it  does  not  follow:  'man  is  an  ani- 
mal, therefore  it  is  true  to  say  that  man  is  an  animal',  although 
the  consequent  is  contained  in  the  antecedent  in  the  exercised 
act.  Granted  further  that  it  follows,  though  not  formally,  I 
say  that  this  other  does  not  follow:  'I  say  that  I  say  what  is 
false,  therefore  in  what  I  say  I  am  simply  truthful',  or  only 
in  a  certain  respect  and  not  simply.  .  .  .  Similarly  it  follows  in 
some  cases:  'What  I  say  is  true,  therefore  I  am  simply  truth- 
ful', as  (e.g.)  here :  'It  is  true  that  I  say  that  man  is  an  animal, 
therefore  I  am  simply  truthful',  viz.  in  those  cases  in  which 
there  is  truth  both  in  the  act  signified  and  the  act  exercised. 
But  in  our  case  (in  proposito)  there  is  falsity  in  the  act  signified 
and  truth  in  the  act  exercised.  It  follows  then :  'It  is  true  that  I 
exercise  the  act  of  speaking  about  what  is  false;  therefore  that 
about  which  I  exercise  it  is  false'. 

A  comparison  of  this  text  with  27.13  shows  that  we  have  here 
almost  exactly  the  modern  distinction  between  use  and  mention.  But 
Pseudo-Scotus,  employing  the  same  terminology,  teaches  just  the 
opposite  to  Burleigh. 

These  two  examples  are  enough  to  show  the  state  of  affairs  in  the 
13th  century.  When  we  come  to  Ockham  the  antinomies  are  no 
longer  dealt  with  in  sophistics,  but  in  a  special  chapter  About 
Insolubles  (35.10).  After  that  such  a  treatise  becomes  an  essential 
part  of  scholastic  logic.  We  pass  over  the  further  stages  of  develop- 
ment, which  are  mostly  not  known,  and  show  how  far  the  matter 
had  got  by  the  time  of  Paul  of  Venice  at  the  end  of  the  Middle  Ages. 



1.  The  Liar 

35.11  I  compose  the  much-disputed  insoluble  by  positing 
(1)  that  Socrates  utters  this  proposition:  'Socrates  says  what 
is  false',  and  this  proposition  is  A,  and  (2)  (that  he)  utters  no 
other  (proposition  besides  A),  (where  the  proposition  A)  (3) 
signifies  so  exactly  and  adequately  that  it  must  not  be  varied 
in  the  present  reply.  That  posited,  I  submits  and  ask  whether 
it  is  true  or  false.  If  it  is  said  that  it  is  true,  contrariwise :  it  is 
consistent  with  the  whole  case  that  there  is  no  other  Socrates 
but  this  Socrates,  and  that  posited,  it  follows  that  A  is  false. 
But  if  it  is  said  that  A  is  false,  contrariwise:  it  is  consistent 
with  the  whole  case  that  there  are  two  Socrateses  of  which 
the  first  says  A,  and  the  second  that  there  is  no  God:  if  that 
is  taken  with  the  statement  of  the  case,  it  follows  that  A  is  true. 

35.12  I  suppose  therefore  that  Socrates,  who  is  every 
Socrates,  utters  this  and  no  other  proposition:  'Socrates  says 
what  is  false',  which  exactly  and  adequately  signifies  (what  it 
says) ;  let  it  be  A.  Which  being  supposed,  it  follows  from  what 
has  been  said  that  A  is  false;  and  Socrates  says  A,  therefore 
Socrates  says  what  is  false.  This  consequence  is  valid,  and  the 
antecedent  is  true,  therefore  also  the  consequent;  but  the 
antecedent  is  A,  therefore  A  is  true. 

Secondly  it  is  argued:  What  is  false  is  said  by  Socrates, 
therefore  Socrates  says  what  is  false.  The  consequence  holds 
from  the  passive  to  its  active.  But  the  antecedent  is  true, 
therefore  also  the  consequent,  and  the  antecedent  is  A, 
therefore  A  is  true.  Since,  however,  the  antecedent  is  true,  it  is 
evident  that  its  adequate  significate  is  true.  But  it  is  a  con- 
tradiction that  it  should  be  true. 

Thirdly  it  is  argued:  the  contradictory  opposite  of  A  is 
false,  therefore  A  is  true.  The  consequence  holds  and  the 
antecedent  is  proved:  for  this:  'no  Socrates  says  what  is 
false'  is  false,  and  this  is  the  contradictory  opposite  of  A ; 
therefore  the  contradictory  opposite  of  A  is  false.  The  conse- 
quence and  the  minor  premiss  hold,  and  I  prove  the  major: 
Since,  A  is  false;  but  a  certain  Socrates  says  A;  therefore  a 
certain  Socrates  says  what  is  false.  Or  thus:  No  Socrates  says 
what  is  false;  therefore  no  Socrates  says  the  false  .4..  The 
consequence  holds  from  the  negative  distributed  super- 
ordinate  to  its  subordinate.  The  consequent  is  false,  therefore 
also  the  antecedent. 



2.  Other  antinomies 

Besides  this  'famous'  insoluble  there  is  a  long  series  of  similar 
antinomies  that  derive  from  it,  of  which  we  give  some  examples 
from  Paul  of  Venice,  omitting  the  always  recurring  words  'Socrates 
who  is  all  Socrates'  and  'which  signify  exactly  as  the  terms  suggest 
(pretendunt)' : 

35.13  Socrates  .  .  .  believes  this:  'Socrates  is  deceived'  .  .  . 
and  no  other  (proposition). 

35.14  Socrates  believes  this  and  no  other:  'Plato  is  deceiv- 
ed' ..  .  but  Plato  .  .  .  believes  this:  'Socrates  is  not  deceived'. 

35.15  Socrates  .  .  .  says  this  and  nothing  else:  'Socrates 

35.16  'Socrates  is  sick';  'Plato  answers  falsely  (male)'; 
'Socrates  will  have  no  penny';  ('Socrates  will  not  cross  the 
bridge')  ;*  where  it  is  supposed  that  every  sick  man,  and  only 
one  such,  says  what  is  false,  and  that  every  well  man,  and 
only  one  such,  says  what  is  true  (and  correspondingly  for  the 
three  other  cases).  .  .  .  On  these  suppositions  I  assert  that 
Socrates  .  .  .  utters  only  the  following:  'Socrates  is  sick'  etc. 

Those  are  the  so-called  'singular  insolubles'.  There  follow  on  them 
the  'quantified'  ones: 

35.17  I  posit  the  case  that  this  proposition  'it  is  false'  is 
every  proposition. 

35.18  Let  this  be  the  case,  that  there  are  only  two  proposi- 
tions, A  and  B,  A  false,  and  B  this:  'A  is  all  that  is  true'. 

35.19  I  posit  that  A,  B  and  C  are  all  the  propositions, 
where  A  and  B  are  true,  and  C  is  this:  'every  proposition  is 
unlike  this'  indicating  A  and  B. 

35.20  I  posit  that  A  and  B  are  all  the  propositions,  where 
A  is  this :  'the  chimera  exists'  .  .  .  and  B  this :  'every  proposi- 
tion is  false'. 

35.21  Let  A,  B  and  C  be  all  the  propositions  .  .  .  where  A  is 
this:  'God  exists',  Bthis:  'man  is  an  ass',  C**  this:  'there  are 
as  many  true  as  false  propositions'. 

35.22  The  answer  to  be  given  would  be  similar  on  the 
supposition  that  there  were  only  five  propositions  ...  of 
which  two  were  true,  two  false,  and  the  fifth  was:  'there  are 
more  false  than  true  (propositions)'. 

*    Inserted  according  to  the  words  just  following. 
*  *   The  text  has  D. 



Then  some  'exponible'  insolubles: 

35.23  I  posit  that  'this  is  the  only  exclusive  proposition' 
is  the  only  exclusive  (proposition).  .  .  . 

35.24  Let  this  be  a  fallacy  about  exceptives:  'no  proposition 
besides  A  is  false',  supposing  that  this  is  A,  and  that  it  is 
every  proposition. 

35.25  I  posit  that  A,  B,  and  C  are  all  the  propositions  .  .  . 
that  A  and  B  are  true,  and  that  C  is  this  exclusive:  'every 
proposition  besides  the  exclusive  is  true'. 

35.26  The  answer  is  similar  ...  on  the  supposition  that 
every  man  besides  Socrates  says:  'God  exists',  and  that 
Socrates  says  only  this:  'every  man  besides  me  says  what  is 

These  are  only  a  few  examples  from  the  rich  store  of  late  scholastic 

1.  The  first  twelve  solutions 

35.27  The  first  opinion  states  that  the  insoluble  is  to  be 
solved  by  reference  to  the  fallacy  of  the  form  of  speech  (11.19). 
.  .  .  And  if  it  is  argued:  'Socrates  utters  this  falsehood,  there- 
fore Socrates  says  what  is  false',  one  denies  the  consequence 
and  says:  'This  is  the  fallacy  of  the  form  of  speech,  because 
by  reason  of  the  (reference  of  the)  speech  the  term  'false' 
supposes  for  'Socrates  etc'  in  the  antecedent,  but  for  some- 
thing else  in  the  consequent.  .  .  . 

35.28  The  second  opinion  solves  the  insolubles  by  the 
fallacy  of  false  cause  (11.24)  .  .  .  since  the  antecedent  seems 
to  be  the  cause  of  the  consequent  but  is  not.  .  .  . 

35.29  The  third  opinion  says  that  when  Socrates  says 
'Socrates  says  what  is  false',  the  word  'says',  although  in  the 
present  tense,  ought  to  be  understood  for  the  time  of  the 
instant  immediately  preceding  the  time  of  utterance.  There- 
fore it  denies  it  (the  proposition),  saying  that  it  is  false.  And 
then  to  the  argument:  'this  is  false  and  Socrates  says  it, 
therefore  Socrates  says  a  false  (proposition)',  they  say  that 
the  verb  'says'  is  verified  for  different  times  in  the  antecedent 
and  consequent.  .  .  . 

35.30  The  fourth  opinion  states  that  nobody  can  say  that  he 
says  what  is  false  or  understand  that  he  understands  what  is 
false,  nor  can  there  be  any  proposition  on  which  an  insoluble 
can  be  based.  This  opinion  is  repugnant  to  sense  and  thought. 



For  everybody  knows  that  a  man  can  open  his  mouth  and 
form  these  utterances:  'I  say  what  is  false'  or  sit  down  and 
read  similar  ones.  .  .  . 

35.31  The  fifth  opinion  states  that  when  Socrates  says 
that  he  himself  says  what  is  false,  he  says  nothing.  .  .  .  This 
opinion  is  likewise  false  because  in  so  saying,  Socrates  says 
letters,  syllables,  dictions  and  orations  as  I  have  elsewhere 
shown.  Further  Socrates  is  heard  to  speak,  therefore  he  says 
something.  Again  they  would  have  to  say  that  if  this,  and  no 
other,  proposition  was  written :  'it  is  false',  that  nothing  would 
be  written,  which  is  evidently  impossible. 

The  fifth  opinion  counts  the  insoluble  as  deprived  of  sense. 

35.32  The  sixth  opinion  states  that  the  insoluble  is  neither 
true  nor  false  but  something  intermediate,  indifferent  to 
each.  They  are  wrong  too,  because  every  proposition  is  true 
or  false,  and  every  insoluble  is  a  proposition,  therefore  every 
insoluble  is  true  or  false.  .  .  . 

That  is  an  effort  to  solve  the  antinomy  in  a  three-valued  logic. 

35.33  The  seventh  opinion  states  that  the  insoluble  is  to 
be  solved  by  reference  to  the  fallacy  of  equivocation.  For  when 
it  is  said:  'Socrates  says  what  is  false'  they  distinguish  about 
the  'saying'  according  to  an  equivocation:  for  it  can  signify 
saying  that  is  exercised  or  that  is  thought  (conceptum).  And 
by  'saying  that  is  exercised'  is  meant  that  which  is  in  course  of 
accomplishment;  it  expresses  the  judgment  and  is  not  com- 
pletely a  dictum.  But  by  'saying  that  is  thought'  is  meant 
(what  happens)  when  a  man  has  said  something  or  spoken  in 
some  way  and  immediately  after  he  says  that  he  says  that, 
or  speaks  in  that  way.  E.g.  supposes  that  Socrates  says 
'God  exists'  and  immediately  after:  'Socrates  says  what  is 
true'.  This  opinion  says  that  when  Socrates  begins  to  say 
'Socrates  says  what  is  false',  if  'saying'  be  taken  for  exercised 
saying,  it  is  true;  but  if  for  saying  in  thought,  it  is  false.  And 
if  it  is  argued:  'nothing  false  is  said  by  Socrates;  and  this  is 
said  by  Socrates;  therefore  it  is  not  false'  -  they  say  that  the 
major  is  verified  for  saying  in  thought,  and  the  minor  for 
exercised  saying,  and  so  (the  argument)  does  not  conclude. 
But  this  solution  is  no  use,  for  let  it  be  supposed  that  the 
speech  is  made  with  exercised  saying,  and  the  usual  deduction 
will  go  through.  .  .  . 



This  solution  corresponds  with  that  of  Pseudo-Scotus  above 

35.34  The  eighth  opinion  states  that  no  insoluble  is  true  or 
false  because  nothing  such  is  a  proposition.  For  although 
every  or  any  insoluble  be  an  indicative  statement  signifying 
according  as  its  signification  is  or  is  not,  yet  this  is  not 
sufficient  for  it  to  be  called  a  'proposition'.  Against  this 
opinion  it  is  argued  that  it  follows  from  it  that  there  are  some 
two  enunciations  of  which  the  adequate  significate  is  one  and 
the  same,  yet  one  is  a  proposition,  the  other  not,  as  is  clear 
when  one  supposes  these:  'this  is  false'  and  'this  is  false', 
indicating  in  both  cases  the  second  of  them.  .  .  . 

This  is  again  a  quite  'modern'  conception.  Paul  of  Venice,  and.  it 
would  seem,  the  majority  of  late  Scholastics,  did  not  like  it. 

35.35  The  ninth  opinion  states  that  the  insoluble  is  true  or 
false,  but  not  true  and  not  false.  .  .  . 

Here  the  alternative  lA  is  true  or  false'  seems  to  be  admitted,  but 
'A  is  true'  and  'A  is  false'  to  be  both  rejected. 

35.36  The  tenth  opinion  solves  the  insoluble  by  reference 
to  the  fallacy  of  in  a  certain  respect  and  simply  (11.24), 
saying  that  an  insoluble  is  a  difficult  paradox  (paralogismus) 
arising  from  (a  confusion  between  what  is)  in  a  certain  respect 
and  simply,  due  to  the  reflection  of  some  act  upon  itself  with 
a  privative  or  negative  qualification.  So  in  solving,  it  says  that 
this  consequence  is  not  valid :  'this  false  thing  is  said  by 
Socrates,  therefore  a  false  thing  is  said  by  Socrates',  supposing 
that  Socrates  says  the  consequent  and  not  something  else 
which  is  not  part  of  it  -  because  the  argument  proceeds  from 
a  certain  respect  to  what  is  simply  so ;  for  the  antecedent  only 
signifies  categorically,  but  the  consequent  hypothetically, 
since  it  signifies  that  it  is  true  and  that  it  is  false.  .  .  . 

35.37  The  eleventh  opinion,  favouring  the  opinion  just 
expounded,  states  that  every  insoluble  proposition  signifies 
that  it  is  true  and  that  it  is  false,  when  understood  as  referring 
to  its  adequate  significate.  For,  as  is  said,  every  categorical 
proposition  signifies  that  that  for  which  the  subject  and 
predicate  suppose  is  or  is  not  the  same  thing,  and  the  being  or 
not  being  the  same  thing  is  for  the  proposition,  affirmative  or 
negative,  to  be  true;  therefore  every  categorical  proposition, 
whether  affirmative  or  negative,  signifies  that  itself  is  true, 



and  every  insoluble  proposition  falsifies  itself;  therefore  every 
insoluble  proposition  signifies  that  it  is  true  or  that  it  is 
false.  .  .  . 

The  last  two  opinions  consider  the  insoluble  to  be  equivalent  to  a 
copulative  proposition.  Why  it  should  be  so  we  shall  see  below  (35.44) . 

35.38  The  twelfth  opinion,  commonly  held  by  all  today,  is 
that  an  insoluble  proposition  is  a  proposition  which  is  supposed 
to  be  mentioned,  and  which,  when  it  signifies  precisely  accord- 
ing to  the  circumstances  supposed,  yields  the  result  that  it  is 
true  and  that  it  is  false.  E.g.  if  a  case  be  posited  about  an 
insoluble,  and  it  is  not  posited  how  that  insoluble  should 
signify,  it  is  to  be  answered  as  though  outside  time:  e.g.  if 
it  be  supposed  that  Socrates  says:  'Socrates  says  what  is 
false'  without  further  determination,  the  proposition  advanc- 
ed: 'Socrates  says  what  is  false'  is  to  be  doubted.  But  if  it 
be  supposed  that  the  insoluble  signifies  as  the  terms  suggest, 
the  supposition  is  admitted  and  the  insoluble  is  granted,  and 
one  says  that  it  is  false.  And  if  it  be  said :  'this  is  false :  "Socra- 
tes says  what  is  false",  therefore  it  signifies  as  it  is  not,  but 
signifies  that  Socrates  says  what  is  false,  therefore  etc'  -  the 
consequence  is  denied.  But  in  the  minor  it  should  be  added 
that  it  signifies  precisely  so,  and  if  that  is  posited,  every 
such  supposition  is  denied.  .  .  . 

The  'time  of  obligation'  here  referred  to  is  a  technical  term  of 
scholastic  discussion  (tractatus  de  obligationibus:  cf.  §  26,  D),  on 
which  very  little  research  has  so  far  been  done.  It  means  the  time 
during  which  the  disputant  is  bound  to  some  (usually  arbitrary) 

2.  The  thirteenth  solution 

35.39  The  thirteenth  opinion  states  a  number  of  conjuncts, 
some  in  the  form  of  theses  (conclusionum),  others  in  the  form 
of  suppositions,  others  in  the  form  of  propositions  or  corol- 
laries ;  but  all  these  can  be  briefly  stated  in  the  form  of  theses 
and  corollaries. 

35.40  The  first  thesis  is  this :  no  created  thing  can  distinctly 
represent  itself  formally,  though  it  can  do  so  objectively. 
This  is  clear,  since  no  created  thing  can  be  the  proper  and 
distinct  formal  cognition  of  itself;  for  if  something  was  to  be 
so,  anything  would  be  so,  since  there  would  be  no  more 
reason  in  one  case  than  in  another.   E.g.  we  say  that  the 



king's  image  signifies  the  king  not  formally  but  objectively, 
while  the  mental  concept  which  we  have  of  the  king  signifies 
the  king  not  objectively  but  formally,  because  it  is  the  formal 
cognition  of  the  king.  But  if  it  be  said  that  it  represents  itself 
distinctly,  this  will  be  objectively,  by  another  concept 
(noiilia)  and  not  formally,  by  itself. 

35.41  Second  thesis:  no  mental  proposition  properly  so- 
called  can  signify  that  itself  is  true  or  that  itself  is  false. 
Proof:  because  otherwise  it  would  follow  that  some  proper 
and  distinct  cognition  would  be  a  formal  cognition  of  itself, 
which  is  against  the  first  thesis. 

From  this  thesis  it  follows  that  the  understanding  cannot 
form  a  universal  mental  proposition  properly  so-called  which 
signifies  that  every  mental  proposition  is  false,  such  as  this 
mental  (proposition):  'every  mental  proposition  is  false', 
understanding  the  subject  to  suppose  for  itself;  nor  can  it 
form  any  mental  proposition  properly  so-called  which  signifies 
that  any  other  is  false  which  in  turn  signifies  that  the  one 
indicated  by  the  first  is  false;  nor  any  mental  proposition 
properly  so-called  which  signifies  that  its  contradictory  is 
true,  as  this  one:  'this  is  true'  indicating  its  contradictory.  .  .  . 

The  last  two  texts  contain  a  notably  acute  formulation  of  the 
veto  on  circulus  vitiosus  (48.21),  and  so  of  the  most  important  modern 
idea  about  the  solution  of  the  antinomies. 

35.42  The  third  thesis  is  this :  a  part  of  a  mental  proposition 
properly  so-called  cannot  suppose  for  that  same  proposition 
of  which  it  is  a  part,  nor  for  the  contradictory  of  that  proposi- 
tion; nor  can  a  part  of  a  proposition  that  signifies  in  an 
arbitrary  way  suppose  for  the  corresponding  mental  proposi- 
tion properly  so-called.  From  which  it  follows  that  if  this 
mental  proposition  is  formed,  and  no  other:  'every  mental 
proposition  is  universal',  it  would  be  false. 

35.43  Fourth  thesis:  there  might  be  a  vocal  or  written  or 
mental  proposition  improperly  so-called  which  had  reflection 
on  itself,  because  all  such  signify  in  an  arbitrary  way  and  not 
naturally,  objectively  but  not  formally.  But  a  mental  proposi- 
tion properly  so-called  is  a  sign  that  represents  naturally  and 
formally,  and  it  is  not  in  our  power  that  such  a  sign  should 
signify  whatever  we  want,  as  it  is  in  the  case  of  a  vocal, 
written  or  mental  sign  improperly  so-called. 

From  this  thesis  it  follows  that  every  insoluble  proposition 



is  a  vocal,  written  or  mental  proposition  improperly  so-called; 
and  a  part  of  any  such  can  suppose  for  the  whole  of  which  it 
is  a  part. 

35.44  The  fifth  thesis  is  this :  to  every  insoluble  proposition 
there  corresponds  a  true  mental  proposition  properly  so- 
called,  and  another  one  properly  so-called,  false.  This  is 
evident  in  the  following:  'this  is  false'  indicating  itself,  which 
corresponds  to  one  such  mental  proposition,  'this  is  false', 
which  is  true.  And  the  second  part  is  proved.  For  this  vocal 
proposition  is  false,  therefore  it  signifies  that  a  mental  one  is 
false,  but  not  the  one  expressed,  therefore  another  one  which 
is  true,  viz.  'this  is  false',  indicating  the  first  mental  one  which 
indicates  a  vocal  or  written  one. 

35.45  From  this  thesis  there  follow  some  corollaries. 
First,  that  any  insoluble  proposition,  and  its  contradictory  too, 
is  a  manifold  proposition  (propositio  plures)  because  there 
correspond  to  it  a  number  of  distinct  (inconiundae)  mental 

Second,  there  are  some  propositions,  vocally  quite  similar 
and  with  terms  supposing  for  the  same  things,  one  of  which 
is  a  manifold  proposition,  but  not  the  other.  This  is  clear  in  the 
following:  'this  is  false'  and  'this  is  false'  where  each  'this' 
indicates  the  second  proposition. 

Third  corollary,  every  insoluble  proposition  is  simultaneous- 
ly true  and  false,  and  its  contradictory  likewise,  because  two 
mental  propositions  of  which  one  is  true  and  the  other  false 
contradict  one  another,  though  neither  is  simply  true  or 
simply  false,  but  in  a  certain  respect.  .  .  . 

3.  The  fourteenth  solution 

35.46  The  fourteenth  opinion,  which  is  the  basis  of  many 
of  the  preceding  ones  and  so  of  those  disputants  who  try 
rather  to  evade  (the  difficulties)  than  to  answer,  states  that 
the  insolubles  are  to  be  solved  by  means  of  the  fallacy  of  the 
accident,  according  to  which  paradoxes  (paralogismi)  arise 
in  two  ways,  by  variation  of  the  middle  term  or  of  one  of  the 
extremes.  By  variation  of  the  middle,  as  when  the  middle 
supposes  for  something  different  in  the  major  to  what  it 
supposes  for  in  the  minor,  and  conversely.  And  similarly 
when  an  extreme  is  varied.  This  opinion  therefore  says  that 
when  Socrates  says  'Socrates  says  what  is  false',  he  says  what 
is  false.  And  then  in  reply  to  the  argument:  'Socrates  says 



this,  and  this  is  false,  therefore  Socrates  says  what  is  false' 
they  deny  the  consequence,  saying  that  here  is  a  fallacy  of  the 
accident  due  to  variation  in  an  extreme;  for  the  term  'false' 
supposes  for  something  in  the  minor  for  which  it  does  not 
suppose  in  the  conclusion.  Similarly  if  it  is  argued  from  the 
opposite  saying  of  Socrates:  'nothing  false  is  said  by  Socrates; 
this  is  false;  therefore  this  is  not  said  by  Socrates',  this  is  a 
fallacy  of  the  accident  due  to  variation  of  the  middle ;  for  the 
term  'false'  supposes  for  something  in  the  major  for  which  it 
does  not  suppose  in  the  minor. 

To  show  that,  they  presuppose  that  in  no  proposition  does 
a  part  suppose  for  the  whole  of  which  it  is  a  part,  nor  is  it 
convertible  with  the  whole,  nor  antecedent  to  the  whole. 
From  which  it  is  clear  that  the  proposition  'Socrates  says 
what  is  false,  signifies  that  Socrates  says  what  is  false,  not, 
however,  the  false  thing  that  he  says,  but  some  false  thing 
distinct  from  that;  but  because  he  only  says  that  proposition, 
therefore  it  is  false.  .  .  . 

This  opinion  has  been  met  with  already  in  Pseudo-Scotus  (35.06; 
and  is  adopted  by  others  too. 

4.  Preliminaries  to  the  solution  of  Paul  of  Venice 

After  expounding  fourteen  opinions  none  of  which  are  acceptable 
to  him,  Paul  of  Venice  gives  his  own  solution,  and  takes  occasion  to 
collect  the  current  late-scholastic  teachings  relevant  to  the  antino- 
mies. We  reproduce  the  essentials. 

35.47  To  explain  the  fifteenth  opinion,  which  I  know  to 
be  that  of  good  (logicians)  of  old  times,  three  chapters  (articuli) 
are  adduced.  The  first  contains  an  explanation  of  terms,  the 
second  introductory  suppositions,  the  third  our  purpose  in  the 
form  of  theses. 

35.48  As  to  the  first,  this  is  the  first  division :  every  insoluble 
arises  either  from  our  activity  or  from  a  property  of  the 
expression  (vocis).  Our  acts  are  twofold,  some  interior,  others 
exterior.  Interior  are  such  as  imagining,  thinking  etc. ;  exterior 
are  bodily  ones  such  as  saying,  speaking  etc.  Insolubles  arising 
from  our  activity  are:  'Socrates  says  what  is  false',  'Socrates 
understands  what  is  false'  etc.  Properties  of  the  expression 
are  such  as  being  subject,  having  appellation,  being  true  or 
false,  being  able  to  be  true,  not  being  true  of  something  other 
than  itself,  and  so  simply  (de  se)  false,  and  not  being  false  of 
itself  or  of  something  else.  And  so  there  arise  from  properties 



of  the  expression  insolubles  like  these:  'it  is  false',  'nothing 
is  true',  'the  proposition  is  not  verified  of  itself.  .  . 

35.49  The  second  division  is  this:  some  propositions  have 
reflection  on  themselves,  some  do  not.  A  proposition  having 
reflection  on  itself  is  one  whose  signification  reflects  on  itself, 
e.g.  'it  is  every  complex  thing',  or  'this  is  false',  indicating 
itself.  A  proposition  without  reflection  on  itself  is  one  whose 
significate  is  not  referred  to  itself,  e.g.  'God  exists'  and  'man 
is  an  ass'. 

35.50  The  third  division  is  this:  of  propositions  having 
reflection  on  themselves  some  have  this  reflection  immediately, 
others  have  it  mediately.  .  .  . 

35.51  The  fourth  division:  of  propositions  having  reflec- 
tions on  themselves,  some  have  the  property  that  their 
significations  terminate  solely  at  themselves,  e.g.  'this  is 
true',  'this  is  false',  indicating  themselves.  But  others  have 
the  property  that  their  significations  terminate  both  at 
themselves  and  at  other  things,  e.g.  'every  proposition  is 
true',  'every  proposition  is  false'.  For  they  do  not  only  signify 
that  they  alone  are  true  or  false,  but  that  other  propositions 
distinct  from  them  are  so  too.  .  .  . 

35.52  It  follows  that  no  proposition  has  reflection  on  itself 
unless  it  contains  a  term  that  is  appropriated  to  signify  the 
proposition,  such  as  are  the  terms  'true',  'false',  'universal', 
'particular',  'affirmative',  'negative',  'to  be  granted',  'to  be 
denied',  'to  be  doubted'  and  so  on.  But  not  every  proposition 
containing  such  a  term  has  to  have  reflection  on  itself,  as  is 
clear  in  these  cases:  'it  is  false',  when  this  is  true,  and  again 
'this  is  true'  indicating  'God  exists';  for  such  does  not  have 
reflection  on  itself,  but  its  signification  is  directed  solely  to 
what  is  indicated  etc.  .  .  . 

There  again  is  the  'modern'  notion  of  the  vicious  circle  (48.21). 
Here  are  a  few  more  preliminaries : 

35.53  The  first  introductory  supposition  is  this:  that  that 
proposition  is  true  whose  adequate  significate  is  true,  and  if 
its  being  true  contains  no  contradiction.  .  .  . 

35.54  Second  supposition:  that  proposition  is  said  to  be 
false  which  falsifies  itself  or  whose  falsity  arises  not  from  the 
terms  but  from  its  false  adequate  significate.  From  which  it 
follows  that  there  is  a  false  proposition  with  a  true  adequate 
significate,  as  is  clear  in  the  following:  'that  is  false',  indicating 



itself.  That  it  is  false  is  evident,  since  it  states  that  it  is  false, 
therefore  it  is  false;  and  so  its  adequate  significate  is  true,  since 
it  is  true  that  it  (the  proposition)  is  false.  It  follows  that 
every  proposition  which  falsifies  itself  is  false,  and  that  not 
every  proposition  which  verifies  itself  is  true ;  since  this : 
'every  proposition  is  true'  verifies  itself  but  is  not  true,  as  is 

35.55  The  third  supposition  is  this:  two  propositions  are 
equivalent  (invicem  convertuntur)  if  their  adequate  significates 
are  identical.  For  let  A  and  B  be  two  such  propositions  having 
the  same  adequate  significate,  and  I  argue  thus:  A  and  B 
have  all  extremes  the  same,  vocally  and  in  writing,  and  in 
thought,  and  similar  copulas,  and  there  is  no  indication 
belonging  to  one  which  does  not  belong  to  the  other;  then 
they  are  equivalent. 

There  follow  some  further  preliminary  suppositions  taken  from 
the  generally  received  teaching  about  supposition  and  consequence. 
Finally  this: 

35.56  The  last  supposition  is  this:  a  part  of  a  proposition 
can  stand  for  the  whole  of  which  it  is  a  part,  as  also  for 
everything  which  belongs  to  it,  without  restriction,  whether 
in  thought  or  in  writing  or  in  speech. 

Thereby  is  rejected  the  thirteenth  opinion  (35.39 ff.),  and  with 
it  the  modern  principle  according  to  which  an  insoluble  is  not  a 
proposition  since  it  contains  a  part  standing  for  the  whole  (48.12 f.). 
This  principle  seems  to  be  presupposed  in  various  ways  by  the 
fourth  (35.30),  fifth  (35.31),  eighth  (35.34),  tenth  (35.36),"  and 
eleventh  (35.37)  opinions. 

The  rejection  of  the  thirteenth  opinion  means  that  the  current 
modern  distinction  of  language  and  meta-language  was  not  adopted 
by  Paul  of  Venice  for  his  own  solution.  But  it  is  explicitly  accepted 
in  the  fifth  thesis  of  the  thirteenth  opinion  (35.44),  more  or  less  so 
in  some  of  the  other  opinions. 

5.  The  solution  of  Paul  of  Venice 

Paul's  own  solution  is  very  like  that  of  the  eleventh  (35.37)  and 
twelfth  (35.38)  opinions,  and  so  we  do  not  reproduce  his  long  and 
difficult  text.  It  consists  essentially  in  a  sharp  distinction  between 
the  ordinary  and  'exact  and  adequate'  meaning  of  the  insoluble 
proposition,  where  'exact  and  adequate'  connotes: 

(1)  the  semantic  correlate,  that  to  which  it  refers; 

(2)  that  the  proposition  itself  is  true. 



This  was  said  already  in  35.57,  though  without  the  use  of  'exact 
and  adequate',  and  with  a  universality  that  Paul  does  not  approve. 
We  repeat  the  main  ideas,  as  they  underlie  his  own  solution.  Some 
simplification  will  be  effected,  and  formalization  used.  The  first 
thing  is  to  set  out  the  antinomy,  for  which  four  extralogical  axioms 
are  employed : 

(1)^4  signifies:  A  is  false. 

(2)  If  A  signifies  p,  then  A  is  true  if  and  only  if  p. 

(3)  If  A  signifies  p,  then  A  is  false  if  and  only  if  not-p. 

(4)  A  is  false  if  and  only  if  A  is  not  true.  (1)  is  the  'insoluble' 
proposition  itself,  (2)-(4)  are  various  formulations  of  the  Aristotelian 
definitions  of  truth  and  falsity  (10.35).  Substituting  lA  is  false' 
for  'p'  in  (2),  we  get  by  (1): 

(5)  A  is  true  if  and  only  if  A  is  false,  which  with  (4)  gives : 

(6)  A  is  true  if  and  only  if  A  is  not  true,  which  in  turn  yields : 

(7)  A  is  not  true, 
and  so  by  (4) : 

(8)  A  is  false. 

But  if  we  put  lA  is  false'  for  'p'  in  (3),  we  get: 

(9)  A  is  false  if  and  only  if  A  is  not  false,  which  immediately 

(10)  A  is  not  false, 

in  contradiction  to  (8).  Here  then  is  a  genuine  antinomy. 

But  the  antinomy  does  not  emerge  if  we  operate  with  the  'exact 
and  adequate'  meaning  instead  of  the  simple  one.  (1)  and  (4)  remain, 
but  the  other  two  axioms  take  on  these  forms : 

(2')  If  A  signifies  p,  then  A  is  true  if  and  only  if  [(1)  A  is  true,  and 
(2)  p]. 

(3')  If  A  signifies  p,  then  A  is  false  if  and  only  if  not  [(1)  A  is  true, 
and  (2)  p],  since  as  has  been  said,  a  proposition  has  'exact  and 
adequate'  signification  when  it  signifies  that  it  is  itself  true,  and  that 
what  it  states  is  as  it  is  stated  to  be. 

The  first  part  of  the  deduction  now  goes  through  analogously  to 
that  given  above,  and  we  again  reach: 

(8')  A  is  false. 
But  putting  'A  is  false'  for  'p'  in  (3')  gives: 

(9')  A  is  false  if  and  only  if  not  [(1)  A  is  true,  and  (2)  A  is  false], 
to  which  we  can  apply  the  de  Morgan  laws  (cf.  31.35)  to  get: 

(10')  A  is  false  if  and  only  if  either  (1)  A  is  not  true,  or  (2)  A  is  not 
false,  i.e.  in  view  of  (4): 

(IT)  A  is  false  if  and  only  if  either  (1)  A  is  false  or  (2)  A  is  not 

As  that  alternation  is  logically  true,  being  a  substitution  in  the 
law  of  excluded  middle  (cf.  31.35),  the  first  part  of  the  equivalence 
must  also  be  true,  giving  us: 

(12')  A  is  false 



which  so  far  from  being  in  contradiction  to  (8')  is  equiform  with  it. 
The  antinomy  is  solved. 

So  far  as  we  know,  the  medieval  logicians  only  treated  of  semantical, 
not  of  logical  antinomies.  But  the  solutions  contain  all  that  is  required 

for  those  as  well. 


In  summary,  we  can  make  the  following  statements  about  medie- 
val formal  logic,  in  spite  of  our  fragmentary  knowledge: 

1.  Scholasticism  created  a  quite  new  variety  of  formal  logic.  The 
essential  difference  between  this  and  the  one  we  found  among  the 
ancients,  consists  in  its  being  an  endeavour  to  abstract  the  laws  and 
rules  of  a  living  (Latin)  language,  regard  had  to  the  whole  realm  of 
semantical  and  syntactical  functions  of  signs. 

2.  This  endeavour  led  to  the  codification  of  a  far-reaching  and 
thorough  semantics  and  syntax;  semiotic  problems  hold  the  fore- 
front of  interest,  and  nearly  all  problems  are  treated  in  relation  to 

3.  Hence  this  logic  is  nearly  entirely  conducted  in  a  meta-language 
(§  26,  B)  with  a  clear  distinction  between  rules  and  laws.  Most  of  the 
theorems  are  thought  of  as  rules  and  formulated  descriptively. 

4.  The  problem  of  logical  form  (§  26,  C)  is  posed  and  solved  with 
great  acumen. 

5.  Problems  of  propositional  logic  and  technique  are  investigated 
as  thoroughly  and  in  as  abstract  a  way  as  anywhere  among  the 

6.  Asserloric  term-logic  consists  here  essentially  in  a  re-interpreta- 
tion and  acute  development  of  syllogistic.  But  there  are  also  other 
kinds  of  problem  in  evidence,  such  as  that  of  plural  quantification,  of 
the  null  class,  perhaps  of  relation-logic,  etc. 

7.  Modal  logic,  both  of  propositions  and  terms,  became  one  of  the 
most  important  fields  of  investigation.  Not  only  was  the  traditional 
system  analysed  with  amazing  thoroughness,  but  quite  new  prob- 
lems were  posed  and  solved,  especially  in  the  propositional  domain. 

8.  Finally  the  problem  of  semantical  antinomies  wras  faced  in  really 
enormous  treatises.  Numerous  antinomies  of  this  kind  were  posited, 
and  we  have  seen  more  than  a  dozen  different  solutions  attempted. 
Between  them  they  contain  nearly  every  essential  feature  of  what  we 
know  today  on  this  subject. 

So  even  in  the  present  incomplete  state  of  knowledge,  we  can 
state  with  safety  that  in  scholastic  formal  logic  we  are  confronted 
with  a  very  original  and  very  fine  variety  of  logic. 



Transitional  Period 


It  is  usual  to  put  the  close  of  the  medieval  period  of  history  at  the 
end  of  the  15th  century.  Of  course  that  does  not  mean  that  typically 
scholastic  ways  of  thought  did  not  persist  longer;  indeed  very  im- 
portant scholastic  schools  arose  in  the  16th  and  17th  centuries  and 
accomplished  deep  and  original  investigations  —  it  is  enough  to 
mention  Cajetan  and  Vittoria.  But  there  was  no  more  research  into 
formal  logic;  at  most  we  find  summaries  of  earlier  results. 

Instead  there  slowly  grew  up  something  quite  new,  the  so-called 
'classical'  logic.  Within  this  extensive  movement  which  held  the 
field  in  hundreds  of  books  of  logic  for  nearly  four  hundred  years,  one 
can  distinguish  three  different  tendencies:  (1)  humanism  (inclusive 
of  those  later  17th  century  thinkers  who  were  humanist  in  their 
approach  to  logic);  it  is  purely  negative,  a  mere  rejection  of  Scho- 
lasticism; (2)  'classical'  logic  in  the  narrower  sense;  (3)  more  recent 
endeavours  to  broaden  the  bounds  of  (2).  Typical  examples  of  the 
three  are  L.  Valla  and  Peter  Ramus,  the  Logique  du  Port  Royal, 
and  W.  Hamilton. 

In  what  follows  we  shall  first  quote  some  passages  to  illustrate 
the  general  attitude  of  authors  of  books  entitled  'Logic'  in  this 
period,  then  some  that  contain  contributions  to  logical  questions, 
whether  scholastic  or  mathematico-logical,  though  these  contribu- 
tions are  of  small  historical  importance. 


Interest  centres  much  more  on  rhetorical,  psychological  and 
epistemological  problems  than  on  logical  ones.  The  humanists,  and 
many  'classical'  logicians  after  them,  expressly  reject  all  formalism. 
That  they  did  not  at  the  same  time  reject  logic  entirely  is  due  to 
their  superstitious  reverence  for  all  ancient  thinkers,  Aristotle  in- 
cluded. But  everything  medieval  was  looked  on  as  sheer  barbarism, 
especially  if  connected  with  formal  logic.  Here  is  an  instance.  Valla 
writes : 

36.01  I  am  often  in  doubt  about  many  authors  of  the  dia- 
lectical art,  whether  to  accuse  them  of  ignorance,  vanity  or 
malice,  or  all  at  once.  For  when  I  consider  the  numerous 
errors  by  which  they  have  deceived  themselves  no  less  than 
others,  I  ascribe  them  to  negligence  or  human  weakness.  But 
when  on  the  other  hand  I  see  that  everything  they  have 
transmitted  to  us  in  endless  books  has  been  given  in  quite  a 
few  rules,  what  other  reason  can  I  suppose  than  sheer  pride? 
In  amusing  themselves  by  letting  the  branches  of  the  vine 


H  U  M  A  N  I  S  M 

spread  far  and  wide,  they  have  changed  the  true  vine  into  a 
wild  one.  And  when  -  this  is  the  worst  -  I  see  the  sophisms, 
quibbles  and  misrepresentations  which  they  use  and  teach, 
I  can  only  kindle  against  them  as  against  people  who  teach  the 
art  of  piracy  rather  than  navigation,  or  to  express  myself 
more  mildly,  knowledge  of  wrestling  instead  of  war. 

And  again,  about  the  third  figure  of  the  syllogism: 

36.02  0  trifling  Polyphemus!  0  peripatetic  family,  that 
loves  trifles!  0  vile  people,  whoever  have  you  heard  arguing 
like  this?  Indeed,  who  among  you  has  ever  presumed  to 
argue  so?  Who  permitted,  endured,  understood  one  who 
argued  thus? 

It  is  on  these  'grounds'  that  the  third  figure  is  to  be  invalid! 
In  another  manner,  but  the  content  goes  deeper,  Descartes  ex- 
presses himself: 

36.03  We  leave  out  of  account  all  the  prescription  of  the 
dialecticians  by  which  they  think  to  rule  human  reason, 
prescribing  certain  forms  of  discourse  which  conclude  so 
necessarily  that  in  relying  on  them  the  reason,  although  to 
some  extent  on  holiday  from  the  informative  and  attentive 
consideration  (of  the  object),  can  yet  draw  some  certain 
conclusion  by  means  of  the  form. 

Evidently  such  an  attitude  will  not  bring  one  to  any  logic.  Peter 
Ramus  holds  a  special  position  among  the  humanists.  Though,  at 
least  in  his  first  period,  he  was  perhaps  the  most  radical  anti- 
Aristotelian,  yet  he  succeeded  in  formulating  on  occasion  some 
interesting  thoughts,  and  published  extensive  treatises  on  formal 
logic.  However,  the  following  gives  some  idea  of  the  general  level 
of  his  logic: 

36.04  Moreover,  two  further  connected  (i.e.  conditional) 
moods  were  added  by  Theophrastus  and  Eudemus,  in  which 
the  antecedent  is  negative  and  the  consequent  affirmative. 
The  third  connected  mood,  then,  takes  the  contradictory  of 
the  antecedent  and  concludes  to  the  contradictory  (of  the 
consequent),  e.g. 

If  the  Trojans  have  come  to  Italy  without  due  permission, 
they  will  be  punished; 
but  they  came  with  permission, 
therefore  they  will  not  be  punished.  .  .  . 



36.05  The  fourth  connected  mood  takes  the  consequent 
and  concludes  to  the  antecedent:  .  .  . 

If  nothing  bad  had  happened,  they  would  be  here  already; 

but  they  are  here, 

therefore  nothing  bad  has  happened. 

36.06  This  mood  is  the  rarest  of  all,  but  natural  and  useful, 
strict  and  correct,  and  it  never  produces  a  false  conclusion 
from  true  premisses.  ...  * 

Of  course  Theophrastus  taught  no  such  moods;  both  theorems 
are  formally  invalid  and  hold  only  by  reason  of  the  matter  in  parti- 
cular cases.  It  is  instructive  to  compare  these  thoughts  with  the 
treatment  of  similar  problems  in  the  Stoics  (22.04 f.)  and  Scholastics 



In  so  bad  a  milieu  logic  could  not  last  long,  yet  there  were  some 
thinkers  among  the  humanists,  Melanchthon  for  instance,  who  without 
being  creative  logicians,  had  a  good  knowledge  of  Aristotle.  It  was 
through  them  that  in  the  17th  century  the  form  of  logic  developed 
which  we  call  the  'classical'  in  the  narrower  sense,  partly  among 
the  so-called  Protestant  Scholastics,  partly  in  Cartesian  circles. 
Perhaps  the  most  important  representative  work  is  the  Logique 
ou  Varl  de  penser  of  P.  Nicole  and  A.  Arnault.  We  describe  the 
contents  of  this  work,  since  they  give  the  best  survey  of  the  problems 
considered  in  'classical'  logic. 

The  book  has  four  parts,  about  ideas,  judgments,  arguments,  and 
method.  In  the  first  part  the  Aristotelian  categories  (ch.  3)  and 
predicables  (ch.  7)  are  briefly  considered,  along  with  some  points  of 
semantics  (Des  idees  des  choses  el  des  idees  des  signes,  ch.  2),  com- 
prehension and  extension  (ch.  6).  The  other  eleven  chapters  are 
devoted  to  epistemological  reflections. 

The  second  part  roughly  corresponds  to  the  content  of  Aristotle's 
Hermeneia,  and  includes  also  considerations  on  definition  and 
division  (ch.  15-16). 

In  the  third  part  the  authors  expound  categorical  syllogistic, 
apparently  following  Peter  of  Spain  and  so  as  a  set  of  rules,  but  with 
use  of  singular  premisses  in  the  manner  of  Ockham  (34.01  f.).  Four 
figures  are  recognized,  with  nineteen  moods  (the  subaltern  moods 
are  missing).  There  follows  a  chapter  on  hypothetical  syllogisms, 
and  (ch.  12)  a  theory  of  syllogismes  conjonctifs  (the  Stoic  compounds) 
using  formulas  of  term-logic,  e.g. 

*   Admittedly  these  moods,  as  Professor  A.  Church  has  stated,  do  not  appear 
in  all  editions.  Perhaps  Ramus  saw  his  own  mistake. 



36.07  If  there  is  a  God,  one  must  love  him: 
But  there  is  a  God ; 

Therefore  one  must  love  him. 

Then  there  are  some  considerations  about  dialectical  loci. 

When  we  compare  this  with  scholastic  logic,  the  main  things 
missing  are  the  doctrines  of  supposition,  consequences,  antinomies, 
and  modal  logic.  The  main  topics  covered  are  those  of  the  Categories, 
Hermeneia,  and  the  first  seven  chapters  of  the  first  hook  of  the 
Prior  Analytics,  but  the  treatment  is  often  scholastic  rather  than 
Aristotelian,  for  which  we  instance  the  use  of  the  mnemonics 
Barbara,  Celarenl  etc.  and  the  metalogical  method  of  exposition. 

The  Logica  Hamburgensis  of  J.  Jungius  (1635)  is  much  better, 
and  richer  in  content;  but  it  did  not  succeed  in  becoming  established. 
The  Logique  ou  Vart  de  penser,  also  called  the  Port  Royal  Logic, 
became  the  standard  text-book,  a  kind  of  Summulae  of  'classical' 
logic.  All  other  text-books  mainly  repeated  its  contents. 


'Classical'  logic  is  characterized  not  only  by  its  poverty  of  con- 
tent but  also  by  its  radical  psychologism.  Jungius  provides  a  good 

36.08  1.  Logic  is  the  art  of  distinguishing  truth  from  falsity 
in  the  operations  of  our  mind  (mentis). 

2.  There  are  three  operations  of  the  mind :  notion  or  concept, 
enunciation,  and  dianoea  or  discourse. 

3.  Notion  is  the  first  operation  of  our  mind,  in  which  we 
express  something  by  an  image;  in  other  words  a  notion  is 
a  simulacrum  by  which  we  represent  things  in  the  mind.  .  .  . 

5.  Enunciation  is  the  second  operation  of  the  mind,  so 
compounded  of  notions  as  to  bring  about  truth  or  falsity. 
E.g.  these  are  true  enunciations:  the  sun  shines,  man  is  a 
biped,  the  oak  is  a  tree.  .  .  . 

9.  It  is  to  be  noted  that  a  notion  and  the  formation  of  a 
notion,  an  enunciation  and  the  effecting  of  an  enunciation, 
an  argumentation  and  the  construction  of  an  argumentation 
are  one  and  the  same. 

This  is  admittedly  an  extreme  case.  But  when  one  thinks  that 
the  text  is  from  Jungius,  one  of  the  best  logicians  of  the  17th 
century,  one  cannot  but  marvel  at  the  extent  to  which  the  under- 
standing of  logic  has  disappeared.  Even  Boole  will  maintain  much 
this  idea  of  logic. 



Poor  in  content,  devoid  of  all  deep  problems,  permeated  with  a 
whole  lot  of  non-logical  philosophical  ideas,  psychologist  in  the 
worst  sense,  -  that  is  now  we  have  to  sum  up  the  'classical'  logic: 

It  may,  however,  be  remarked  that  A.  Menne  (Logik  und  Existenz, 
131,  note  34)  has  propounded  a  distinction  between  an  at  least 
relatively  pure  'classical'  logic,  and  a  'traditional'  philosophical  and 
psychological  logic,  though  these  terms  are  commonly  used  syno- 
nymously. Of  the  former,  J.  N.  Keynes  (1906)  may  be  taken  as  in 
every  sense  the  best  representative,  but  W.  E.  Johnson  (1921) 
shows  how  relative  is  the  distinction. 


Formed  by  this  logic  and  its  prejudices,  modern  philosophers  such 
as  Spinoza,  the  British  empiricists,  Wolff,  Kant,  Hegel  etc.  could 
have  no  interest  for  the  historian  of  formal  logic.  When  compared 
with  the  logicians  of  the  4th  century  B.C.,  the  13th  and  20th  cen- 
turies a.d.  they  were  simply  ignorant  of  what  pertains  to  logic  and 
for  the  most  part  only  knew  what  they  found  in  the  Port  Royal 

But  there  is  one  exception,  Leibniz  (1646-1716).  So  far  from  being 
an  ignoramus,  he  was  one  of  the  greatest  logicians  of  all  time,  which 
is  the  more  remarkable  in  that  his  historical  knowledge  was  rather 
limited.  His  place  in  the  history  of  logic  is  unique.  On  the  one  hand 
his  achievement  constitutes  a  peak  in  the  treatment  of  a  part  of  the 
Aristotelian  syllogistic,  where  he  introduced  many  new,  or  newly 
developed  features,  such  as  the  completion  of  the  combinatorial 
method,  the  exact  working  out  of  various  methods  of  reduction, 
the  method  of  substitution,  the  so-called  'Eulerian'  diagrams,  etc. 
On  the  other  hand  he  is  the  founder  of  mathematical  logic. 

The  reason  why  Leibniz  is,  nevertheless,  named  in  this  section, 
and  only  named,  is  that  his  great  achievements  in  the  realm  of 
mathematical  logic  are  little  relevant  to  the  history  of  problems, 
since  they  remained  for  long  unpublished  and  were  first  discovered 
at  the  end  of  the  19th  century  when  the  problems  he  had  dealt 
with  had  already  been  raised  independently. 

Only  in  one  respect  does  he  seem  to  have  exercised  a  decisive 
influence,  in  forming  the  idea  of  mathematical  logic.  The  pertinent 
passages  will  be  quoted  incidentally  in  the  next  section.  Here  we 
limit  ourselves  to  quoting  some  of  his  contributions  to  syllogistic 
theory,  and  showing  some  of  his  diagrams. 


The  idea  underlying  the  distinction  between  comprehension  and 
extension  is  a  very  old  one:  it  is  presupposed,  for  instance,  in  the 
Isagoge  of  Porphyry  (24.02  ff.);  the  scholastic  doctrine  of  supposition 



has  a  counterpart  of  it  in  the  theory  of  simple  (27.17 IT. )  and  personal 
(27.24 fT.)  supposition  with  an  elaborate  terminology.  But  the 
expressions  comprehension  and  elendue  are  first  found  in  the  Port 
Royal  Logic.  Leibniz  evidently  has  the  idea,  but  without  an  establish- 
ed terminology. 

We  first  cite  an  extract  from  his  article  De  formae  logicae  com- 
probatione  per  linearum  ductus: 

36.09  Up  to  now  we  have  assessed  the  quantities  of  terms 
in  respect  of  (ex)  the  individuals.  And  when  it  was  said: 
'every  man  is  an  animal',  it  was  meant  (consider alum  est) 
that  all  human  individuals  form  a  part  of  the  individuals 
that  fall  under  'animal'  (esse  partem  individuorum  animalis). 
But  in  respect  of  (secundum)  ideas,  the  assessment  proceeds 
just  conversely.  For  while  men  are  a  part  of  the  animals, 
conversely  the  notion  of  animal  is  a  part  of  the  notion  apply- 
ing to  man,  since  man  is  a  rational  animal. 

So  Leibniz  had  a  fairly  accurate  idea  of  comprehension  and 
extension,  as  well  as  of  their  inter-relationship.  Now  we  come  to  the 
Port  Royal  Logic: 

36.10  Now  in  these  universal  ideas  (idees)  there  are  two 
things  (choses)  which  it  is  important  to  keep  quite  distinct: 
comprehension  and  extension. 

I  call  the  comprehension  of  an  idea  the  attributes  which 
it  contains  and  which  cannot  be  taken  away  from  it  without 
destroying  it;  thus  the  comprehension  of  the  idea  of  triangle 
includes  extension,  figure,  three  lines,  three  angles,  the 
equality  of  these  three  angles  to  two  right-angles,  etc. 

I  call  the  extension  of  an  idea  the  subjects  to  which  it  applies, 
which  are  also  called  the  inferiors  of  a  universal  term,  that 
being  called  superior  to  them.  Thus  the  idea  of  triangle  in 
general  extends  to  all  different  kinds  of  triangle. 


Already  in  his  youthful  work  De  arte  combinatoria  Leibniz  resumed 
the  thought  of  Albalag  (32.25  ff.),  without  being  acquainted  with  it, 
and  proved  that  there  is  a  fourth  figure  of  assertoric  syllogism 
(36.11).  Later,  he  gave  a  complete  and  correct  table  of  the  twenty- 
four  syllogistic  moods,  in  which  he  deduced  the  moods  of  the  second 
and  third  figures  from  those  of  the  first,  using  the  first  reduction 
procedure  of  Aristotle  (§  14,  D).  We  give  a  table  which  reproduces 
the  deduction  of  the  second  and  third  figure  moods,  this  time  in  the 
original  language.  ' Regressus'  means  contraposition. 




Barbara  primae 

A  CD 



Barbara  primae 














Hinc  Baroco 

Hinc  Bocardo 









Celarent  primae 




Celarent  primae 














Hinc  Festino 

Hinc  Disamis 









Darii  primae 




Darii  primae 














Hinc  Camestres 

Hinc  Ferison 









Ferio  primae 




Ferio  primae 














Hinc  Cesare 

Hinc  Datisi 









Barbari  primae 




Barbari  primae 














Hinc  Camestres 

Hinc  Felapton 









Celaro  primae 




Celaro  primae 














Hinc  Cesaro 

Hinc  Darapti 









(cf.  13.21,  13.22. 


The  idea  of  representing  class  relations  and  syllogistic  moods  by 
geometrical  figures  was  familiar  to  the  ancient  commentator 
(24.34) ;  how  far  it  was  current  among  the  Scholastics  is  not  yet 
known.  The  use  of  circles  is  usually  ascribed  to  L.  Euler  (1701-83) 
(cf.  his  Lettres  a  une  princesse  d'Allemagne,  1768),  while  that  of 
straight  lines  is  associated  with  the  name  of  Lambert.  But  the 
former  are  to  be  found  earlier  in  J.  C.  Sturm  (1661)  (36.13),  and 
the  latter  in  Alstedius  (1614)  (36.14).  Schroder  (36.15)  notes  that 
L.  Vives  was  using  angles  and  triangles  in  1555.*   Leibniz's  use 

For  the  foregoing  and  some  further  details  vid.  A.  Menne  (36.16) 


p  ■ 




'■'7»l  <>>***.  Me--  ♦;■ 

9~*$ifi* -{  d3  -4    I  j 

&»^     r -U-LJ 

•     L    1>-    V-  •>  ...  ,*C 

LI'  0  a 


1  ~3i- 







Syllogistic  diagrams  bv  Leilmiz    3G. 



Y  Z 



of  circles  and  other  diagrammatic  methods  remained  unpublished 
till  1903.  We  reproduce  a  page  of  his  MS  (36.17)  which  contains  both 
circular  and  rectilinear  diagrams. 

Such  methods  of  presentation  were  much  considered  and  further 
developed  from  the  time  of  Euler  onwards.  J.  Venn  (1860)  intro- 
duced ellipses  for  his  investigations  of  the  relations  between  more 
than  three  classes,  and  marked  with  a  star  every  region  representing 
a  non-void  class.  Three  of  his  diagrams  are  reproduced  in  (36.18) 
and  a  systematic  development  is  considered  by  W.  E.  Hocking 

A  different  kind  of  diagram,  mnemonic  rather  than  expository 
of  probative,  due  to  Johnson,  may  be  added  here. 

36.20  The  attached  diagram,  taking  the  place  of  the  mnemo- 
nic verses,  indicates  which  moods  are  valid,  and  which  are 
common  to  different  figures.  The  squares  are  so  arranged  that 
the  rules  for  the  first,  second  and  third  figures  also  show  the 
compartments  into  which  each  mood  is  to  be  placed,  according 
as  its  major,  minor  or  conclusion  is  universal  or  particular, 
affirmative  or  negative.  The  valid  moods  of  the  fourth  figure 
occupy  the  central  horizontal  line. 

In  the  figure,  the  superscripts  V  and  V  indicate  the  propositions 
that  may  be  weakened  or  strengthened  by  subalternation. 


While  all  points  so  far  referred  to  fall  within  the  general  scheme 
of  Aristotelian  logic,  Bentham's  doctrine  of  the  quantification  of 
the  predicate,  usually  ascribed  to  Hamilton,  is  directly  opposed  to 
Aristotle's  teaching  (12.03).  At  the  same  time,  as  can  be  seen  from 
the  texts,  it  is  a  development  of  the  scholastic  doctrine  of  exponibles. 
It  has  this  historical  importance,  that  it  shows  the  kind  of  problem 
being  considered  by  logicians  at  the  time  of  Boole,  and  in  some 
degree  throws  light  on  the  origin  of  Boole's  calculus. 



We  give  two  texts  from  G.  Bentham  (1827)  first,  then  one  from 
Hamilton  (1860): 

36.21  In  the  case  where  both  terms  of  a  proposition  are 
collective  entities,  identity  and  diversity  may  have  place: 

1.  Between  any  individual  referred  to  by  one  term,  and  any 
individual  referred  to  by  the  other.  Ex. :  The  identity  between 
equiangular  and  equilateral  triangles. 

2.  Between  any  individual  referred  to  by  one  term,  and 
any  one  of  a  part  only  of  the  individuals  referred  to  by  the 
other.  Ex. :  The  identity  between  men  and  animals. 

3.  Between  any  one  of  a  pari  only  of  the  individuals  referred 
to  by  one,  and  any  one  of  a  pari  only  of  the  individuals  referred 
to  by  the  other  term.  Ex. :  The  identity  between  quadrupeds 
and  swimming  animals. 

36.22  Simple  propositions,  considered  in  regard  to  the  above 
relations,  may  therefore  be  either  affirmative  or  negative; 
and  each  term  may  be  either  universal  or  partial.  These 
propositions  are  therefore  reducible  to  the  eight  following 
forms,  in  which,  in  order  to  abstract  every  idea  not  connected 
with  the  substance  of  each  species,  I  have  expressed  the  two 
terms  by  the  letters  X  and  Y,  their  identity  by  the  mathe- 
matical sign  =,  diversity  by  the  sign  ||  ,  universality  by  the 
words  in  toto,  and  partiality  by  the  words  ex  parte;  or,  for 
the  sake  of  still  further  brevity,  by  prefixing  the  letters  I  and 
p,  as  signs  of  universality  and  partiality.  These  forms  are, 


X  in  toto      = 

Y  ex  parte 

or  IX 

=  pY 


X  in  toto 

Y  ex  parte 

or  IX 

II    PY 


X  in  toto      = 

Y  in  toto 

or  IX 

=  IY 


X  in  toto 

Y  in  toto 

or  IX 

1    IF 


X  ex  parte   = 

Y  ex  parte 

or  pX 

=  PY 


X  ex  parte 

Y  ex  parte 

or  pX 

1    PY 


X  ex  parte   = 

Y  in  toto 

or  pX 

=  tY 


X  ex  parte    \\ 

Y  in  toto 

or  pX 

II    tY 

Hamilton  writes: 

36.23  The  second  cardinal  error  of  the  logicians  is  the  not 
considering  that  the  predicate  has  always  a  quantity  in 
thought,  as  much  as  the  subject;  although  this  quantity  be 
frequently  not  explicitly  enounced,  as  unnecessary  in  the 
common  employment  of  language ;  for  the  determining  notion 
or  predicate  being  always  thought  as  at  least  adequate  to,  or 



coextensive  with,  the  subject  or  determined  notion,  it  is 
seldom  necessary  to  express  this,  and  language  tends  ever 
to  elide  what  may  safely  be  omitted.  But  this  necessity 
recurs,  the  moment  that,  by  conversion,  the  predicate  becomes 
the  subject  of  the  proposition;  and  to  omit  its  formal  state- 
ment is  to  degrade  Logic  from  the  science  of  the  necessities 
of  thought,  to  an  idle  subsidiary  of  the  ambiguities  of  speech. 
An  unbiased  consideration  of  the  subject  will,  I  am  confident, 
convince  you  that  this  view  is  correct. 

1°,  That  the  predicate  is  as  extensive  as  the  subject  is 
easily  shown.  Take  the  proposition,  -  'All  animal  is  man',  or, 
'All  animals  are  men'.  This  we  are  conscious  is  absurd.  .  .  . 
We  feel  it  to  be  equally  absurd  as  if  we  said,  -  'All  man  is  all 
animal',  or  ,'A11  men  are  all  animals'.  Here  we  are  aware  that 
the  subject  and  predicate  cannot  be  made  coextensive.  If  we 
would  get  rid  of  the  absurdity,  we  must  bring  the  two  notions 
into  coextension,  by  restricting  the  wider.  If  we  say  -  'Man 
is  animal',  [Homo  est  animal],  we  think,  though  we  do  not 
overtly  enounce  it,  'All  man  is  animal'.  And  what  do  we  mean 
here  by  animal?  We  do  not  think,  all,  but  some,  animal.  And 
then  we  can  make  this  indifferently  either  subject  or  predi- 
cate. We  can  think,  -  we  can  say,  'Some  animal  is  man', 
that  is,  some  or  all  man;  and,  e  converso,  'Man  (some  or  all)  is 
animal',  viz.  some  animal.  .  .  . 

2°,  But,  in  fact,  ordinary  language  quantifies  the  predicate 
so  often  as  this  determination  becomes  of  the  smallest  import. 
This  it  does  either  directly,  by  adding  all,  some,  or  their 
equivalent  predesignations  to  the  predicate ;  or  it  accomplishes 
the  same  end  indirectly,  in  an  exceptive  or  limitative  form. 

Hamilton  then  proceeds  to  repeat,  in  dependence  on  the  works 
of  various  17th  and  18th  century  logicians,  the  scholastic  doctrine 
of  the  exponibilia  (§  34,  C). 



The  mathematical  variety  of  Logic 


I.  General  Foundations 


The  development  of  the  mathematical  variety  of  logic  is  not  yet 
complete,  and  discussions  still  go  on  about  its  characteristic  scope 
and  even  about  its  name.  It  was  simultaneously  called  'mathemati- 
cal logic',  'symbolic  logic'  and  'logistic'  by  L.  Couturat,  Itelson  and 
Lalande  in  1901,  and  is  sometimes  simply  called  'theoretical  logic'. 
Even  apart  from  the  philosophical  discussions  as  to  whether  or 
how  far  it  is  distinct  from  mathematics,  there  is  no  unanimity  about 
the  specific  characteristics  which  distinguish  it  from  other  forms  of 

However,  there  exists  a  class  of  writings  which  are  generally 
recognized  as  pertaining  to 'mathematical  logic'  ('logistic', 'symbolic 
logic'  etc.).  Analysis  of  their  contents  shows  that  they  are  predomi- 
nantly distinguished  from  all  other  varieties  of  logic  by  two  inter- 
dependent characteristics. 

(1)  First,  a  calculus,  i.e.  a  formalistic  method,  is  always  in  evi- 
dence, consisting  essentially  in  the  fact  that  the  rules  of  operation 
refer  to  the  shape  and  not  the  sense  of  the  symbols,  just  as  in  mathe- 
matics. Of  course  formalism  had  already  been  employed  at  times 
in  other  varieties  of  logic,  in  Scholasticism  especially,  but  it  is  now 
erected  into  a  general  principle  of  logical  method. 

(2)  Connected  with  that  is  a  deeper  and  more  revolutionary 
innovation.  All  the  other  varieties  of  logic  known  to  us  make  use 
of  an  abslractiue  method ;  the  logical  theorems  are  gained  by  abstrac- 
tion from  ordinary  language.  Mathematical  logicians  proceed  in 
just  the  opposite  way,  first  constructing  purely  formal  systems,  and 
later  looking  for  an  interpretation  in  every-day  speech.  This  process 
is  not  indeed  always  quite  purely  applied;  and  it  would  not  be 
impossible  to  find  something  corresponding  to  it  elsewhere.  But  at 
least  since  Boole,  the  principle  of  such  construction  is  consciously 
and  openly  laid  down,  and  holds  sway  throughout  the  realm  of 
mathematical  logic. 

Those  are  the  essential  features  of  mathematical  logic.  Two  more 
should  be  added: 

(3)  The  laws  are  formulated  in  an  artificial  language,  and  consist 
of  symbols  which  resemble  those  of  mathematics  (in  the  narrower 
sense).  The  new  feature  here  is  that  even  the  constants  are  expressed 
in  artificial  symbols;  variables,  as  we  have  seen,  have  been  in  use 
since  the  time  of  Aristotle. 

(4)  Finally,  until  about  1930  mathematical  logic  formulated  its 



theorems  in  an  object  language,  in  this  unlike  the  Scholastics,  but 
in  conformity  with  the  ancients.  That  this  is  no  essential  feature 
is  shown  by  more  recent  developments  and  the  spread  of  metalogical 
formulation.  But  till  1930  the  use  of  the  object  language  is  charac- 

It  may  be  further  remarked  that  it  can  be  said  of  mathematical 
logic,  what  was  finally  said  about  scholastic,  that  it  is  very  rich 
and  very  formalistic.  In  wealth  of  formulae  indeed,  it  seems  to 
exceed  all  other  forms  of  logic.  It  is  also  purely  formal,  being  sharply 
distinguished  from  the  decadent  'classical'  logic  by  its  avoidance  of 
psychological,  epistemological  and  metaphysical  questions. 


G.  W.  Leibniz  generally  ranks  as  the  original  mathematical 
logician,  but  if  he  cannot  count  as  the  founder  of  mathematical 
logic  it  is  because  his  logical  works  were  for  the  most  part  published 
long  after  his  death  (the  essentials  by  L.  Couturat  in  1901).  However, 
he  had  some  successors,  the  most  important  of  whom  were  the 
brothers  Bernoulli  (1685),  G.  Plouquet  (1763,  1766),  J.  H.  Lambert 
(1765,  1782),  G.  J.  von  Holland  (1764),  G.  F.  Castillon  (1803)  and 
J.  D.  Gergonne  (1816/17).*  But  no  school  arose. 

One  who  did  found  a  school,  and  who  stands  at  the  beginning  of 
the  continuous  development  of  mathematical  logic,  is  George 
Boole,  whose  first  pioneer  work,  The  Mathematical  Analysis  of 
Logic,  appeared  in  1847.  In  the  same  year  Augustus  de  Morgan 
published  his  Formal  Logic.  Boole's  ideas  were  taken  further  in 
different  directions  by  R.  L.  Ellis  (1863),  W.  S.  Jevons  (1864), 
R.  Grassmann  (1872),  J.  Venn  (1880,  1881),  Hugh  McColl  (1877/78), 
finally  and  chiefly  by  E.  Schroder  (1877,  1891-95). 

Contemporaneous  with  the  last-named  are  the  works  of  a  new 
group  of  mathematical  logicians  whose  chief  representatives  are 
C.  S.  Peirce  (1867,  1870),  Gottlob  Frege  (1879),  and  G.  Peano 
(1888).  Of  these  three  important  thinkers  only  Peano  founded  a 
considerable  school;  Peirce  and  Frege  went  practically  unnoticed. 
It  was  Bertrand  Russell  (1903)  who  discovered  the  thought  of 
Frege  and  together  with  A.  N.  Whitehead  combined  it  with  his 
own  discoveries  in  Principia  Mathematica  (1910-13),  in  which  the 
symbolism  of  Peano  was  used. 

D.  Hilbert  (1904)  and  L.  E.  J.  Brouwer  (1907,  1908)  were  active 
before  the  appearance  of  the  Principia.  J.  Lukasiewicz  published 
his  first  work  in  this  field  in  1910,  St.  Lesniewski  in  1911.  They 
were  followed  by  A.  Tarski  (1921),  R.  Carnap  (1927),  A.  Heyting 
(1929)  and  K.  Godel  (1930). 

*  Figures  in  parentheses  give  the  year  of  publication  of  the  main  work, 
then  of  the  first  subsequent  important  one. 



These  are  only  a  few  of  the  great  number  of  mathematical  logi- 
cians, which  by  now  is  beyond  count. 


Among  all  these  logicians,  Gottlob  Frege  holds  a  unique  place. 
His  Begriffsschrift  can  only  be  compared  with  one  other  work  in 
the  whole  history  of  logic,  the  Prior  Analytics  of  Aristotle.  The  two 
cannot  quite  be  put  on  a  level,  for  Aristotle  was  the  very  founder  of 
logic,  while  Frege  could  as  a  result  only  develop  it.  But  there  is  a 
great  likeness  between  these  two  gifted  works.  The  Begriffsschrift, 
like  the  Prior  Analytics,  contains  a  long  series  of  quite  new  insights, 
e.g.  Frege  formulates  for  the  first  time  the  sharp  distinction  between 
variables  and  constants,  the  concepts  of  logical  function,  of  a  many- 
place  function,  of  the  quantifier;  he  has  a  notably  more  accurate 
understanding  of  the  Aristotelian  theory  of  an  axiomatic  system, 
distinguishes  clearly  between  laws  and  rules,  and  introduces  an 
equally  sharp  distinction  between  language  and  meta-language, 
though  without  using  these  terms;  he  is  the  author  of  the  theory  of 
description;  without  having  discovered,  indeed,  the  notion  of  a 
value,  he  is  the  first  to  have  elaborated  it  systematically.  And  that 
is  far  from  being  all. 

At  the  same  time,  and  just  like  Aristotle,  he  presents  nearly  all 
these  new  ideas  and  intuitions  in  an  exemplarily  clear  and  systematic 
way.  Already  in  the  Begriffsschrift  we  have  a  long  series  of  mathe- 
matico-logical  theorems  derived  from  a  few  axioms  'without 
interruption'  (luckenlos),  as  Frege  says,  for  the  first  time  in  history. 
Various  other  mathematical  logicians  at  the  same  time,  or  even 
earlier,  expounded  similar  ideas  and  theories,  but  none  of  them  had 
the  gift  of  presenting  all  at  once  so  many,  often  quite  original, 
innovations  in  so  perfect  a  form. 

It  is  a  remarkable  fact  that  this  logician  of  them  all  had  to  wait 
twenty  years  before  he  was  at  all  noticed,  and  another  twenty  before 
his  full  strictness  of  procedure  was  resumed  by  Lukasiewicz.  In  this 
last  respect,  everything  published  between  1879  and  1921  fell  below 
the  standard  of  Frege,  and  it  is  seldom  attained  even  today.  The 
fate  of  Frege's  work  was  in  part  determined  by  his  symbolism. 
It  is  not  true  that  it  is  particularly  difficult  to  read,  as  the  reader 
can  assure  himself  from  the  examples  given  below;  but  it  is  certainly 
too  original,  and  contrary  to  the  age-old  habits  of  mankind,  to  be 

All  that  we  have  said  does  not  mean  that  Frege  is  the  only  great 
logician  of  the  period  now  under  consideration.  We  also  have  to 
recognize  as  important  the  basic  intuitions  of  Boole,  and  many 
discoveries  of  Peirce  and  Peano,  to  name  only  these  three.  The  very 
fact  that  Frege  was  a  contemporary  of  Peirce  and  Peano  forbids 



one  to  treat  him   as  another  Aristotle.   But  of  all   mathematical 
logicians  he  is  undoubtedly  the  most  important. 


The  history  of  mathematical  logic  can  be  divided  into  four  periods. 

1.  Prehistory:  from  Leibniz  to  1847.  In  this  period  the  notion  of 
mathematical  logic  arose,  and  many  points  of  detail  were  formulated, 
especially  by  Leibniz.  But  there  was  no  school  at  this  time,  and 
the  continuous  development  had  not  yet  begun.  There  were,  rather, 
isolated  efforts  which  went  unnoticed. 

2.  The  Boolean  period,  from  Boole's  Analysis  to  Schroder's  Vor- 
lesungen  (vol.  I,  1895).  During  this  period  there  is  a  continuous 
development  of  the  first  form  of  mathematical  logic.  This  form  is  prin- 
cipally distinguished  from  later  ones  in  that  its  practitioners  did  not 
make  the  methods  of  mathematics  their  object  of  study,  but  con- 
tented themselves  with  simply  applying  them  to  logic. 

3.  The  period  of  Frege,  from  his  Begriffsschrift  (1879)  to  the 
Principia  Mathematica  of  Whitehead  and  Russell  (1910-13).  Frege, 
and  contemporaneously  Peirbe  and  Peano,  set  a  new  goal,  to  find 
foundations  for  mathematics.  A  series  of  important  logical  ideas 
and  methods  were  developed.  The  period  reaches  its  peak  with  the 
Principia  which  both  closes  the  preceding  line  of  development  and 
is  the  starting  point  of  a  new  one,  its  fruitfulness  being  due  in  the 
first  place  to  a  thorough  consideration  and  solution  of  the  problem 
of  the  antinomies  which  had  been  a  burning  question  since  the  end 
of  the  19th  century  and  had  not  previously  found  a  solution  in  the 
new  period. 

4.  The  most  recent  period :  since  the  Principia,  and  still  in  progress. 
This  period  can  be  sub-divided:  the  years  from  1910  to  1930  are 
distinguished  by  the  rise  of  metalogic,  finitist  in  Hilbert,  not  so 
in  Lowenheim  and  Skolem;  after  about  1930  metalogic  is  systema- 
tized in  a  formalistic  way,  and  we  have  Tarski's  methodology, 
Carnap's  syntax,  and  the  semantics  of  Godel  and  Tarski  in  which 
logic  and  metalogic  are  combined.  The  'natural'  logics  of  Gentzen 
and  Jaskowski  (1934)  also  belong  here. 

So  we  can  say  that  the  advance  of  metalogic  is  distinctive  of  the 
time  since  1910,  though  new  logical  systems  (in  the  object  language) 
continue  to  appear:  that  of  Lewis  (1918),  the  many-valued  systems  of 
Post  and  Lukasiewicz  (1920-21),  the  intuitionistic  logic  of  Heyting 
(1930).  Finally  the  very  original  systems  of  combinatorial  logic 
by  Schonfinkel  (1924),  Curry  (1930),  Kleene  (1934),  Rosser  (1935) 
and  Church  (1936-41). 

This  fourth  period  will  be  touched  on  only  very  lightly,  in  some 
of  its  problems. 

The  following  table  gives  an  easy  view  of  the  temporal  sequence 



of  the  logicians  we  have  named.  But  it  is  to  be  noticed  (1)  that 
temporal  succession  does  not  always  reflect  actual  influence;  this 
will  be  discussed  more  in  detail  in  the  various  chapters.  (2)  the  subject 
developed  so  fast  after  1870  that  dates  of  births  and  deaths  are 
little  to  the  purpose ;  we  have  preferred  to  give  those  of  publication 
of  the  chief  logical  works. 

G.  W.  v.  Leibniz 

A.  De  Morgan  1847 

C.  S.  Peirce  1867-1870 

G.  Frege  1879 
G.  Peano  1888 

G.  Boole  1847 

R.  C.  Ellis  1863 

W.  S.  Jevons  1864 

R.  Grassmann  1872 

H.  McColl  1877/78 

E.  Schroder  1877 

D.  Hilbert  1904 
L.  Brouwer  1907/8 

B.  Russell  1903 
Principia  1910-1913 

J.  Lukasiewicz  1910 
St.  Lesniewski  1911 
A.  Tarski  1921 
R.  Carnap  1927 
A.  Heyting  1929 
K.  Godel  1930 


Mathematical  logic  is  the  best  known  form  of  logic,  since  many 
of  its  basic  works,  especially  the  Principia,  so  far  from  being  past 
history  are  still  in  current  use.  Then  again  there  have  already  been 
a  number  of  historical  studies  of  the  period.  Among  these  are  the 
work  of  B.  Jourdain  (37.01),  the  historical  sections  of  the  works  of 
C.  I.  Lewis  (37.02)  and  J.  Jorgensen  (37.03).  The  treatise  of  H.  Hermes 
and  H.  Scholz  (37.04)  is  remarkably  rich  in  historical  information. 
Since  1936  we  have  had  as  unique  tools  of  research,  a  biblio- 
graphy of  mathematical  logic  from  Leibniz  to  1935,  and  the  Journal 
of  Symbolic  Logic  containing  a  current  bibliography  and  excellent 
indexes.  Both  are  as  good  as  bibliograj  hy  can  be,  under  the  editor- 
ship of  A.  Church  who  sees  to  them  with  exemplary  punctuality  and 
regularity.  Among  other  contributions  to  the  history  of  this  period 
the  numerous  papers  of  R.  Feys  should  be  mentioned. 



But  still  we  do  not  know  all  about  the  period.  L.  Couturat's 
thorough  and  serious  monograph  on  Leibniz  needs  completing  on 
many  points  in  the  light  of  more  recent  systematic  and  historical 
research;  there  are  also  various  other  treatises  on  Leibniz's  logic. 
Boole,  too,  has  been  fairly  thoroughly  investigated  in  recent  years. 
But  as  yet  there  is  no  detailed  treatment  of  Leibniz's  successors,  no 
monograph  on  Peirce,  above  all  no  thorough  work  on  Frege's  logic, 
without  mentioning  other  less  important  logicians. 


For  the  reason  stated  in  the  introduction,  we  have  tried  to  present 
the  essential  range  of  problems  discussed  in  mathematical  logic  by 
means  of  texts  containing  little  or  no  artificial  symbols.  This  has 
proved  feasible  by  and  large,  but  not  without  exception ;  in  particular, 
at  least  the  basic  methods  have  to  be  explained  in  terms  of  the 
contemporary  symbolism,  e.g.  of  Frege  or  the  Principia.  Then  again, 
we  have  given  the  most  important  theorems  in  the  various  fields 
in  symbolic  formulation,  in  order  to  facilitate  comparison  with 
similar  theorems  developed  in  other  periods. 

The  question  of  what  time-limit  to  put  is  very  difficult,  and  the 
various  periods  within  the  main  one  dovetail  into  each  other  in  such 
a  way  as  to  make  the  drawing  of  sharp  boundaries  impossible.  We 
have  finally  decided  to  close  the  exposition  with  the  Principia, 
touching  lightly  on  a  few  later  developments  which  are  either 
closely  connected  with  matters  discussed  before  1910,  or  of  special 
interest  on  their  own  account.  That  section  (§  49)  is  accordingly  in 
the  nature  of  an  appendix. 

The  reader  will  only  be  able  to  appreciate  the  textual  fragments  that 
follow  if,  first,  he  is  well  acquainted  with  the  fundamental  concepts 
of  contemporary  mathematical  logic  (cf.  §  5,  B);  second,  he  is  able  to 
abstract  from  the  philosophy  (ontology,  epistemology,  psychology 
etc.)  of  the  various  logicians.  For  never  before  have  formal  logicians 
been  so  divided  by  mutually  opposed  philosophies  as  here.  We  need 
only  instance  Frege's  outspoken  Platonism,  and  Boole's  nominalism 
and  even  psychologism.  But  they  have  all  developed  essentially  the 
same  formal  logic. 

That  is  not  to  say  that  the  individual  philosophic  views  have  been 
entirely  without  influence  on  the  form  of  this  or  that  system.  But 
such  influence  has  been  much  slighter  than  an  unbiased  observer 
might  at  first  suppose.  That  the  systems  present  such  different 
appearances  is  due  mainly  to  differences  of  immediate  purpose 
(one  may  compare  Boole  with  Frege,  or  Peano  with  Lukasiewicz), 
and  to  differences  in  the  degree  of  exactness  which  are  more  marked 
here  than  in  any  other  period. 



Two  essentially  distinct  methodological  ideas  seem  to  underlie 
mathematical  logic.  On  the  one  hand  it  is  a  logic  that  uses  a  calculus. 
This  was  developed  in  connection  with  mathematics,  which  at  first 
was  considered  as  the  ideal  to  which  logic  should  approach.  On  the 
other  hand  mathematical  logic  is  distinguished  by  the  idea  of  exact 
proof.  In  this  respect  it  is  no  hanger-on  of  mathematics,  and  this  is 
not  its  model;  it  is  rather  the  aim  of  logic  to  investigate  the  founda- 
tions and  conduct  of  mathematics  by  means  of  more  exact  methods 
than  have  been  customary  among  'pure'  mathematicians,  and  to 
offer  to  mathematics  the  ideal  of  strict  proof. 

In  both  respects  the  name  'mathematical  logic'  is  justified, 
though  for  opposite  reasons;  first,  because  the  new  logic  is  a  result  of 
mathematics,  then  because  it  seeks  to  provide  a  basis  for  that 
science.  But  it  would  be  a  misunderstanding  to  conclude  that  mathe- 
matical logicians  want  to  confine  themselves  to  the  consideration  of 
quantities;  their  aim  from  the  start  has  rather  been  to  construct  a 
quite  general  logic. 

In  what  follows  we  illustrate  both  aspects  with  a  series  of  texts 
which  resume  the  development  of  mathematical  logic. 

1.  Lull 

The  idea  of  a  mechanical  process  to  facilitate  inference  is  already 
present  in  the  combinatorial  arguments  of  the  ancient  Commentators, 
the  Arabs  and  the  Scholastics.  We  have  given  one  example  above 
(32.34),  but  of  course  it  was  only  a  matter  of  determining  correct 
syllogistic  moods.  Raymond  Lull  (1235-1315)  is  the  first  to  lay  claim 
to  a  quite  general  mechanical  procedure.  It  appears  from  the  work 
of  this  remarkable  man  that  he  believed  himself  to  have  found  a 
method  which  permits  one  to  draw  every  kind  of  conclusion  by 
means  of  a  system  of  concentric,  circular  sheets  or  rings,  of  various 
sizes  and  mutually  adjustable,  with  letters  inscribed  on  their  rims. 
Unfortunately  Lull  does  not  express  the  main  ideas  of  this  procedure 
at  all  clearly.  However,  it  will  be  well  to  give  at  least  a  few  passages 
from  his  Ars  Magna,  since  his  doctrine  is  not  only  one  of  the  greatest 
curiosities  in  the  history  of  logic,  but  also  had  some  influence  on 

38.01  The  understanding  longs  and  strives  for  a  universal 
science  of  all  sciences,  with  universal  principles  in  which  the 
principle  of  the  other,  more  special  sciences  would  be  implicit 
and  contained  as  is  the  particular  in  the  universal.  .  .  . 



38.02  This  art  is  divided  into  thirteen  parts,  viz.  the 
alphabet,  the  figures,  definitions,  rules,  tables  .  .  .  (etc.). 

The  alphabet  of  this  art  is  the  following: 

B  signifies  goodness,  difference,  whether,  God,  justice, 

C  signifies  quantity,  conformity,  what,  angel,  prudence, 

We  spare  the  reader  the  further  enumeration  of  this  alphabet. 
But  we  print  a  picture  of  the  'first  figure'  and  here  is  part  of  the 
accompanying  commentary: 

38.03  There  are  four  figures,  as  appears  from  this  page. 
The  first  figure  is  signified  by  A;  and  it  is  circular,  subdivided 
into  nine  compartments.  In  the  first  compartment  is  B,  in 
the  second  C,  etc.  And  it  is  said  to  be  cruciform,  in  that  the 
subject  is  turned  into  the  predicate  and  conversely,  as  when 
one  says:  great  goodness,  good  greatness ;  eternal  greatness, 
great  eternity;  God  the  good  (Deus  bonus),  the  good  God 
(bonus  Deus),  and  correspondingly  for  other  (terms).  By 
means  of  rotations  of  this  kind  the  practitioner  (artista)  can 
see  what  is  converted  and  what  is  not  converted,  such  as 
'God  is  good'  and  the  like,  which  can  be  converted.  But  God 
and  angel  will  not  be  converted,  nor  goodness  and  angel, 
nor  its  goodness  and  (its)  greatness,  and  so  on  with  the  other 

This  text  is  far  from  clear,  and  its  consequences  no  clearer;  it  has, 
moreover,  little  relevance  for  genuine  logic.  But  the  mere  idea  of  such 
a  mechanical  process  was  a  fascinating  one  for  many  people  in  the 
16th  and  17th  centuries. 

2.  Hobbes 

Lull's  ideas  are  to  be  found  expressed  in  an  extreme  form  three 
hundred  years  later,  by  Thomas  Hobbes  (1655).  He  made  no  attempt 
to  carry  them  out,  for  like  most  modern  philosophers  Hobbes  was  no 

38.04  By  ratiocination  I  mean  computation.  Now  to 
compute,  is  either  to  collect  the  sum  of  many  things  that 
are  added  together,  or  to  know  what  remains  when  one  thing 
is  taken  out  of  another.  Batiocination,  therefore,  is  the  same 
with  addition  and  substruction  (sic);  and  if  any  man  add 
multiplication  and  division,  I  will  not  be  against  it,  seeing 
multiplication    is    nothing    but    addition    of    equals    one    to 



another,  and  division  nothing  but  a  substraction  of  equals 
one  from  another,  as  often  as  is  possible.  So  that  all  ratio- 
cination is  comprehended  in  these  two  operations  of  the  mind, 
addition  and  substraction. 

This  is,  to  be  sure,  rather  the  jeu  d' esprit  of  a  dilettante  than  a 
theory  of  mathematical  logic;  no  inference  can  be  interpreted  in  this 
way,  and  Hobbes  never  once  tried  to  do  it.  The  passage  shows  the 
mark  of  his  extreme  verbalism,  inference  being  a  mere  accumulation 
of  words.  However,  this  text  is  historically  important  as  having 
exercised  some  influence  on  Leibniz,  and  it  is  also  characteristic 
of  the  mathematicism  which  largely  dominated  the  new  form  of 
logic  until  Jevons.  But  perhaps  no  logician  was  so  badly  infected 
with  it  as  Hobbes. 

3.  Leibniz 

Leibniz  had  read  Lull  (38.05)  and  cites  Hobbes  too  (38.06).  But  he 
has  much  more  to  offer  than  either  ol  them.  Like  Lull,  he  is  con- 
cerned with  a  universal  basis  for  all  sciences;  like  Lull  again,  his 
basic  philosophy  leads  him  to  think  of  a  purely  combinatorial 
method.  But  this  is  now  to  take  the  form  of  a  calculus,  such  as  is 
employed  in  mathematics ;  logic  is  to  be  thought  of  as  a  generalized 
mathematics.  Leibniz's  most  characteristic  texts  on  this  point  are 
the  following: 

38.07  As  I  was  keenly  occupied  with  this  study,  I  happened 
unexpectedly  upon  this  remarkable  idea,  that  an  alphabet 
of  human  thought  could  be  devised,  and  that  everything 
could  be  discovered  (inveniri)  and  distinguished  (dijudicari) 
by  the  combination  of  the  letters  of  this  alphabet  and  by  the 
analysis  of  the  resulting  words. 

38.08  The  true  method  should  afford  us  a  filum  Ariadnes, 
i.e.  a  sensibly  perceptible  and  concrete  means  to  guide  the 
mind,  like  the  lines  drawn  in  geometry  and  the  forms  of  the 
operations  which  are  prescribed  to  learners  in  arithmetic. 
Without  this  our  mind  could  traverse  no  path  without  going 

38.09  To  discover  and  prove  truths,  the  analysis  of  ideas 
is  necessary,  .  .  .  which  corresponds  to  the  analysis  of  (written) 
characters.  .  .  .  Hence  we  can  make  the  analysis  of  ideas 
sensibly  perceptible  and  conduct  it  as  with  a  mechanical 
thread;  since  the  analysis  of  the  characters  is  something 
sensibly  perceptible. 


The  "first  figure"  of  Lull's  "Ars  Magna".  Cf.  38.03 


38.10  A  characteristic  of  reason,  by  means  of  which  truths 
would  become  available  to  reason  by  some  method  of  calcula- 
tion, as  in  arithmetic  and  algebra,  so  in  every  other  domain, 
so  long  as  it  submits  to  the  course  of  deduction. 

38.11  Then,  in  case  of  a  difference  of  opinion,  no  discussion 
between  two  philosophers  will  be  any  longer  necessary,  as 
(it  is  not)  between  two  calculators.  It  will  rather  be  enough 
for  them  to  take  pen  in  hand,  set  themselves  to  the  abacus, 
and  (if  it  so  pleases,  at  the  invitation  of  a  friend)  say  to  one 
another:  Calculemus! 

38.12  Ordinary  languages,  though  mostly  helpful  for  the 
inferences  of  thought,  are  yet  subject  to  countless  ambiguities 
and  cannot  do  the  task  of  a  calculus,  which  is  to  expose 
mistakes  in  inference  owing  to  the  forms  and  structures  of 
words,  as  solecisms  and  barbarisms.  This  remarkable  advan- 
tage is  afforded  up  to  date  only  by  the  symbols  (notae)  of 
arithmeticians  and  algebraists,  for  whom  inference  consists 
only  in  the  use  of  characters,  and  a  mistake  in  thought  and 
in  the  calculus  is  identical. 

38.13  Hence  it  seems  that  algebra  and  the  mathesis  uni- 
versalis ought  not  to  be  confused  with  one  another.  If  indeed 
mathesis  was  to  deal  only  with  quantity,  or  with  equals  and 
unequals,  or  with  mathematical  ratio  and  proportion,  there 
would  be  nothing  to  prevent  algebra  (which  considers  quan- 
tity in  general)  from  being  considered  as  their  common  part. 
But  mathesis  seems  to  underlie  whatever  the  power  of  imagina- 
tion underlies,  insofar  as  that  is  accurately  conceived,  and  so 
it  pertains  to  it  to  treat  not  only  of  quantity  but  also  of  the 
arrangement  (dispositio)  of  things.  Thus  mathesis  universalis, 
if  I  am  not  mistaken,  has  two  parts,  the  ars  combinatoria 
concerned  with  the  variety  of  things  and  their  forms  or 
qualities  in  general  insofar  as  they  are  subject  to  exact 
inference,  and  the  equal  and  the  unequal;  then  logistic  or 
algebra,  which  is  about  quantity  in  general. 

There  are  here  two  different,  though  connected  ideas:  that  of  an 
'alphabet  of  thought'  and  that  of  mathesis  universalis.  According 
to  the  first,  one  is  to  assign  a  symbol  to  every  simple  idea  and  solve 
all  problems  by  combinations  of  these  symbols.  This  is  very  con- 
sonant with  Leibniz's  philosophy,  in  particular  with  his  doctrine  of 
the  strictly  analytic  character  of  all  necessary  propositions  and  of 
inference  as  a  combining  of  elements.  This  philosophical  view, 
questionable  in  itself,  was  yet  fruitful  for  logic  in  that  it  led  Leibniz 



to  the  notion  of  an  artificial  language  (38.12)  which,  by  contrast  to 
ordinary  languages,  would  be  free  from  ambiguities.  Therein  Leibniz 
is  the  founder  of  symbolic  logic  as  such,  i.e.  of  the  use  of  artificial 
symbols  even  for  logical  constants  (and  not  only  for  variables  as  in 
all  earlier  forms  of  logic). 

The  other  idea  is  that  of  mathesis  universalis  (38.10),  i.e.  of  the 
application  of  calculation  to  all  inferences,  not  only  to  those  that 
are  mathematical  in  the  narrower  sense.  Leibniz  does  not  advocate 
any  mathematicism  such  as  that  of  Hobbes:  malhesis  universalis  is 
sharply  distinguished  from  ordinary  algebra  (here  strangely  called 
'logistic')  and  set  in  contrast  to  it  (38.13).  It  is  only  the  method  that 
is  to  be  applied  to  logic,  and  this  is  not  any  'addition  and  subtrac- 
tion' as  with  Hobbes,  but  simply  formal  operating  with  symbols.  Of 
course  the  idea  of  a  strictly  formalistic  logic,  of  constructing  some 
meaningless  system  which  is  only  interpreted  later,  such  as  we  find  in 
Boole,  is  not  yet  present.  The  calculus  is  to  be  a  filum  Ariadnes  to 
assist  the  mind.  The  process  envisaged  is  therefore  basically  the 
same  as  in  the  earlier  logical  tradition;  formal  laws  are  abstracted 
from  meaningful  sentences.  But  the  principle  of  a  formal  process,  i.e. 
of  calculation,  is  here  clearly  expressed  for  the  first  time,  so  far  as  we 
know.  In  this,  Leibniz  is  the  founder  of  mathematical  logic. 

4.  Lambert 

Some  further  development  of  Leibniz's  ideas  is  to  be  found  in 
Lambert  (1728-1777): 

38.14  Let  us  see,  then,  how  a  more  universal  idea  can  be 
abstracted  from  the  arithmetical  and  algebraic  calculi.  First, 
the  idea  of  quantity  must  be  got  rid  of,  as  being  too  special. 
You  may  put  in  its  place  qualities,  affections,  thinys,  truths, 
ideas  and  whatever  can  be  discussed,  combined,  connected, 
separated  and  chanyed  into  ever  new  forms;  all  and  each  of  these 
substitutions  can  be  made  in  accordance  with  the  nature  of 
things.  For  each  of  these  operations  and  changes,  with  due 
differences  allowed  for,  are  applied  to  quantities. 

Further,  for  the  ideas  of  equality,  equation,  ratio,  relation, 
proportion,  proyression,  etc.  which  occur  in  arithmetic,  more 
universal  ones  are  to  be  substituted.  So  that  in  place  of 
equality  it  will  be  convenient  to  introduce  identity,  in  place  of 
equation  identification,  if  this  word  is  taken  in  its  active  sense, 
in  place  of  proportion  analoyy.  And  if  the  words  relation, 
proyression  be  retained,  their  meaning  is  to  be  extended,  as 
ordinary  usage  suggests,  so  that  they  can  be  thought  of  as 
relations     or     progressions    between     the     thinys,     qualities, 



affections,  ideas  or  truths  to  which  the  calculus  is  to  be  suited. 
And  this  is  to  be  chiefly  noted,  that  those  relations  contribute 
no  little  to  determining  the  form  of  the  calculus,  and  that  all 
those  operations  which  the  object  of  the  calculus  admits  rest  chiefly 
on  them. 

5.  Gergonne 

Gergonne  (1816/17)  comes  much  closer  to  the  idea  of  formalism: 

38.15  It  is  constantly  being  said  that  reasoning  must  only 
be  about  objects  of  which  one  has  a  perfectly  clear  idea,  yet 
often  nothing  is  more  false.  One  reasons,  in  practice,  with 
words,  just  as  one  calculates  with  letters  in  algebra;  and  in 
the  same  way  that  an  algebraic  calculation  can  be  carried  out 
exactly  without  one  having  the  slightest  idea  about  the  sig- 
nification of  the  symbols  on  which  one  is  operating,  in  the 
same  way  it  is  possible  to  follow  a  course  of  reasoning  without 
any  knowledge  of  the  signification  of  the  terms  in  which  it  is 
expressed,  or  without  adverting  to  it  if  one  knows  it.  .  .  .  It  is 
evident,  for  example,  that  one  does  not  have  to  know  the 
nature  of  the  terms  of  a  proposition  in  order  to  deduce  its 
converse  or  subaltern  when  it  admits  of  such.  Doubtless  one 
cannot  dispense  with  a  good  knowledge  of  notions  which  are  to 
be  the  immediate  matter  of  judgment;  but  that  is  quite 
unnecessary  for  concluding  to  a  judgment  from  a  number  of 
others  already  known  to  be  correct. 

This  text  is  not  altogether  clear;  Gergonne  seems  to  equate  the 
(Aristotelian)  use  of  variables  with  formalism.  But  we  can  see  the 
idea  of  formalism  becoming  clearer. 

Gergonne  also  gave  expression  to  an  idea  which  is  not  without 
relevance  to  the  symbolism  of  mathematical  logic: 

38.16  There  is  no  known  language  in  which  a  proposition 
exactly  and  exclusively  expresses  in  which  of  our  five  relations 
both  its  component  terms  stand;  such  a  language  would  have 
five  kinds  of  proposition  and  its  dialectic  would  be  quite 
different  from  that  of  our  languages. 

He  is  referring  there  to  five  relationships  between  the  extensions 
of  two  terms  (or  classes)  which  will  be  spoken  of  later  (40.12).  * 

*  We  learned  of  this  passage  from  a  work  of  J.  A.  Faris. 


6.  Boole 

We  can  find  a  clear  idea  of  formalism,  developed  in  an  exemplary 
way,  in  the  introduction  to  George  Boole's  epoch-making  The 
Mathematical  Analysis  of  Logic  (1847),  in  this  superior  to  much  later 
works,  e.g.  the  Principia. 

38.17  They  who  are  acquainted  with  the  present  state  of 
the  theory  of  Symbolical  Algebra,  are  aware,  that  the  validity 
of  the  processes  of  analysis  does  not  depend  upon  the  inter- 
pretation of  the  symbols  which  are  employed,  but  solely  upon 
the  laws  of  their  combination.  Every  system  of  interpretation 
which  does  not  affect  the  truth  of  the  relations  supposed,  is 
equally  admissible,  and  it  is  thus  that  the  same  process  may, 
under  one  scheme  of  interpretation,  represent  the  solution  of  a 
question  on  the  properties  of  numbers,  under  another,  that 
of  a  geometrical  problem,  and  under  a  third,  that  of  a  problem 
of  dynamics  or  optics.  This  principle  is  indeed  of  fundamental 
importance;  and  it  may  with  safety  be  affirmed,  that  the 
recent  advances  of  pure  analysis  have  been  much  assisted  by 
the  influence  which  it  has  exerted  in  directing  the  current  of 

But  the  full  recognition  of  the  consequences  of  this  impor- 
tant doctrine  has  been,  in  some  measure,  retarded  by  acciden- 
tal circumstances.  It  has  happened  in  every  known  form  of 
analysis,  that  the  elements  to  be  determined  have  been  con- 
ceived as  measurable  by  comparison  with  some  fixed  standard. 
The  predominant  idea  has  been  that  of  magnitude,  or  more 
strictly,  of  numerical  ratio.  The  expression  of  magnitude,  or 
of  operations  upon  magnitude,  has  been  the  express  object 
for  which  the  symbols  of  Analysis  have  been  invented,  and 
for  which  their  laws  have  been  investigated.  Thus  the  abstrac- 
tions of  the  modern  Analysis,  not  less  than  the  ostensive 
diagrams  of  the  ancient  Geometry,  have  encouraged  the 
notion,  that  Mathematics  are  essentially,  as  well  as  actually, 
the  Science  of  Magnitude. 

The  consideration  of  that  view  which  has  already  been 
stated,  as  embodying  the  true  principle  of  the  Algebra  of 
Symbols,  would,  however,  lead  us  to  infer  that  this  conclusion 
is  by  no  means  necessary.  If  every  existing  interpretation  is 
shewn  to  involve  the  idea  of  magnitude,  it  is  only  by  induction 
that  we  can  assert  that  no  other  interpretation  is  possible. 
And  it  may  be  doubted  wither  our  experience  is  sufficient  to 



render  such  an  induction  legitimate.  The  history  of  pure 
Analysis  is,  it  may  be  said,  too  recent  to  permit  us  to  set 
limits  to  the  extent  of  its  applications.  Should  we  grant  to  the 
inference  a  high  degree  of  probability,  we  might  still,  and  with 
reason,  maintain  the  sufficiency  of  the  definition  to  which  the 
principle  already  stated  would  lead  us.  We  might  justly 
assign  it  as  the  definitive  character  of  a  true  Calculus,  that  it  is 
a  method  resting  upon  the  employment  of  Symbols,  whose 
laws  of  combination  are  known  and  general,  and  whose 
results  admit  of  a  consistent  interpretation.  That  to  the 
existing  forms  of  Analysis  a  quantitative  interpretation  is 
assigned,  is  the  result  of  the  circumstances  by  which  those 
forms  were  determined,  and  is  not  to  be  construed  into  a 
universal  condition  of  Analysis.  It  is  upon  the  foundation  of 
this  general  principle,  that  I  purpose  to  establish  the  Calculus  of 
Logic,  and  that  I  claim  for  it  a  place  among  the  acknowledged 
forms  of  Mathematical  Analysis,  regardless  that  in  its  object 
and  in  its  instruments  it  must  at  present  stand  alone. 
From  that  Boole  draws  the  explicit  conclusion: 

38.18  On  the  principle  of  a  true  classification,  we  ought  no 
longer  to  associate  Logic  and  Metaphysics,  but  Logic  and 
Mathematics.  .  .  .  The  mental  discipline  which  is  afforded  by 
the  study  of  Logic,  as  an  exact  science,  is  in  species,  the  same 
as  that  afforded  by  the  study  of  Analysis. 

Leibniz  and  Lambert  had  already  wanted  to  apply  calculation 
to  logic,  and  had  used  the  idea  of  non-quantitative  calculation.  The 
epoch-making  feature  of  Boole's  text  is  the  exemplarily  clear  account 
of  the  essence  of  calculation,  viz.  formalism,  a  process  of  which  the 
'validity  does  not  depend  upon  the  interpretation  of  the  symbols 
which  are  employed,  but  solely  upon  the  laws  of  their  combination'. 
Boole  is  also  aware  of  the  possibility  of  interpreting  the  same  formal 
system  in  different  ways.  This  suggests  that  he  did  not  think  of  logic 
as  an  abstraction  from  actual  processes,  as  all  previous  logicians  had 
done,  but  as  a  formal  construction  for  which  an  interpretation  is 
sought  only  subsequently.  That  is  quite  new,  and  in  contrast  with 
the  whole  tradition,  Leibniz  included. 

7 '.  Peirce 

Finally  we  submit  a  text  from  Peirce's  review  of  Schroder's  logic 
(1896),  which  contains  one  of  the  best  statements  of  the  advantage 
to  be  looked  for  in  a  logical  calculus. 

38.19  It  is  a  remarkable  historical  fact  that  there  is  a 
branch  of  science  in  which  there  has  never  been  a  prolonged 



dispute  concerning  the  proper  objects  of  that  science.  It  is 
mathematics.  Mistakes  in  mathematics  occur  not  infrequently, 
and  not  being  detected  give  rise  to  false  doctrine,  which  may 
continue  a  long  time.  Thus,  a  mistake  in  the  evaluation  of  a 
definite  integral  by  Laplace,  in  his  Mecanique  celeste,  led  to  an 
erroneous  doctrine  about  the  motion  of  the  moon  which 
remained  undetected  for  nearly  half  a  century.  But  after  the 
question  had  once  been  raised,  all  dispute  was  brought  to  a 
close  within  a  year.  .  .  . 

38.20  Hence,  we  homely  thinkers  believe  that,  considering 
the  immense  amount  of  disputation  there  has  always  been 
concerning  the  doctrines  of  logic,  and  especially  concerning 
those  which  would  otherwise  be  applicable  to  settle  disputes 
concerning  the  accuracy  of  reasonings  in  metaphysics,  the 
safest  way  is  to  appeal  for  our  logical  principles  to  the  science 
of  mathematics,  where  error  can  only  long  go  unexploded  on 
condition  of  its  not  being  suspected.  .  .  . 

38.21  Exact  logic  will  be  that  doctrine  of  the  conditions  of 
establishment  of  stable  belief  which  rests  upon  perfectly 
undoubted  observations  and  upon  mathematical,  that  is, 
upon  diagrammatical,  or  iconic,  thought.  We,  who  are  sectaries 
of  'exact'  logic,  and  of  'exact'  philosophy,  in  general,  main- 
tain that  those  who  follow  such  methods  will,  so  far  as  they 
follow  them,  escape  all  error  except  such  as  will  be  speedily 
corrected  after  it  is  once  suspected. 


1.  Bolzano 

A  noteworthy  precursor  of  modern  proof-theory  is  Bernard 

38.22  If  we  now  state  that  M,  N,  0,  .  .  .  are  deducible  from 
A,  B,  C,  .  .  .  and  this  in  respect  of  the  notions  i,  /,  .  .  .:  we  are 
basically  saying,  according  to  what  has  been  said  in  §  155, 
the  following:  'All  ideal  contents  which  in  the  place  of  i,  /,  .  .  . 
in  the  propositions  A,  B,  C,  .  .  .  M,  N,  0,  .  .  .  simultaneously 
verify  the  propositions  A,  B,  C,  .  .  .  has  the  property  of  also 
simultaneously  verifying  the  propositions  M,  TV,  0.  .  .  .'  The 

*   Professor  Hans  Hermes  drew  our  attention  to  this  passage. 

The  rows  of  dots  after  the  groups  of  letters  are  here  part  of  the  text. 



most  usual  way  of  giving  expression  to  such  propositions  is  of 
course:  '//  A,  B,  C,  .  .  .  are  true:  then  also  M,  TV,  0,  .  .  .  are 
true.'  But  we  often  also  say:  'M,  TV,  0,  .  .  .  follow,  or  are  Reduc- 
ible, or  can  be  inferred  from  A,  B,  C,  .  .  .  etc'  In  respect 
of  the  notions  i,  /',...  which  we  consider  as  the  variables  in 
these  propositions,  the  same  remark  is  applicable  as  in  No.  1. 
But  since  according  to  §  155  No.  20  it  is  not  at  all  the  <  •;)-«■ 
with  the  relation  of  deducibility,  as  (it  is)  with  the  relations  of 
mere  compatibility,  that  a  given  content  of  propositions 
A,  B,  C,  ....  on  the  one  hand,  and  M,  TV,  0,  ...  on  the  other, 
can  come  into  this  relationship  merely  because  we  determine 
arbitrarily  which  notions  therein  are  to  count  as  variables: 
it  is  thus  a  rather  startling  statement  when  we  say  that  certain 
propositions  M,  N,  0,  .  .  .  can  be  brought  into  a  relationship  of 
deducibility  with  other  propositions  A,  B,  C,  ...  by  merely 
taking  the  notions  pertaining  to  them  as  variable.  But  in  such 
a  judgment  we  only  say  that  there  are  certain  parts  of  the 
propositions  A,  B,  C,  .  .  .  M,  TV,  0,  .  .  .  which  can  be  considered 
as  variable,  with  the  result  that  every  ideal  content  which  in 
the  place  of  i,  /,  .  .  .  makes  all  of  A,  B,  C,  .  .  .  true,  also  makes 
all  of  M,  TV,  0,  .  .  .  true.  And  thus  we  can  easily  see  from 
§  137  how  such  a  proposition  must  be  expressed  to  bring  out 
its  logically  constant  parts.  'The  notion  of  some  parts  of 
A,  B,  C,  .  .  .  M,  TV,  0,  ...  so  constituted  that  every  arbitrary 
ideal  content  which  in  their  place  verifies  A,  B,  C,  .  .  .  always 
also  verifies  M,  TV,  0,  .  .  .  has  objectivity'.  In  ordinary  speech 
propositions  of  this  kind  are  expressed  just  like  the  preceding 
ones.  It  is  only  from  other  circumstances,  e.g.  from  the 
context,  that  one  can  guess  whether  the  speaker  has  in  his 
mind  determinate  notions  in  respect  to  which  the  retation  of 
deducibility  is  to  be  present,  or  whether  he  only  inlends  to 
intimate  that  there  are  such  notions.  Thus,  e.g.,  it  is  easy 
enough  to  gather  from  the  following  proposition:  'if  Caius  is  a 
man,  and  all  men  are  mortal,  then  Caius,  too,  is  mortal',  that 
it  is  here  intended  to  state  the  deducibility  of  the  proposition : 
Caius  is  mortal,  from  the  two  propositions :  Caius  is  a  man,  and, 
all  men  are  mortal,  in  respect  of  the  three  notions;  Caius,  man 
and  mortal.  This  next  utterance  on  the  other  hand:  'If  in  all 
men  there  stirs  an  irresistible  desire  for  permanence ;  if,  too,  the 
most  virtuous  must  feel  unhappy  at  the  thought  that  he  is 
one  day  to  cease;  then  we  are  not  wrong  to  expect  of  God's 
infinite  goodness  that  he  will  not  annihilate  us  in  death'  - 



this  would  be  subject  to  the  reproach  of  extreme  obscurity, 
since  its  sense  is  not  that  the  said  propositions  stand  in  a 
relationship  of  deducibility  when  some  of  their  notions  (which 
still  have  to  be  ascertained)  have  been  taken  as  variable. 
By  such  an  utterance  it  is  only  intended  to  state  that  notions 
are  present  such  as  to  warrant  inference  from  the  truth  of  the 
antecedent  to  the  truth  of  the  consequent;  but  it  does  not  as 
yet  tell  one  which  these  notions  properly  are. 

2.  Frege 

While  that  text  of  Bolzano's  contains  important  ideas  about  the 
concept  of  deduction  or  deducibility,  the  modern  development  of  this 
second  aspect  of  mathematical  logic  begins  with  Frege.  We  take  the 
essential  texts  from  the  Grundgesetzen  der  Arilhmeiik  (1893);  but  it 
can  easily  be  shown  that  most  of  what  is  said  in  them  was  already 
known  to  this  great  logician  by  1879. 

38.23  In  my  Grundlagen  der  Arithmetik  I  have  tried  to 
make  it  plausible  that  arithmetic  is  a  branch  of  logic  and  does 
not  need  to  take  its  grounds  of  proof  either  from  experience  or 
intuition.  This  will  now  be  confirmed  in  the  present  book,  by 
the  fact  that  the  simplest  laws  of  numbers  can  be  deduced  by 
logical  means  alone.  But  at  the  same  time  this  shows  that 
considerably  higher  demands  must  be  made  on  the  process  of 
proof  than  is  usual  in  arithmetic.  A  region  of  some  ways  of 
inference  and  deduction  must  be  previously  delimited,  and  no 
step  may  be  made  which  is  not  in  accordance  with  one  of 
these.  In  the  passage,  therefore,  to  a  new  judgment,  one  must 
not  be  satisfied  with  the  fact  that  it  is  evidently  correct,  as 
mathematicians  nearly  always  have  been  up  to  now,  but  one 
must  analyze  it  into  its  simple  logical  steps,  which  are  often  by 
no  means  a  few.  No  presupposition  may  remain  unremarked ; 
every  axiom  which  is  needed  must  be  discovered.  It  is  just  the 
tacit  presuppositions,  that  are  made  without  clear  conscious- 
ness, which  obscure  understanding  of  the  epistemological  cha- 
racter of  a  law. 

38.24  The  ideal  of  a  strict  scientific  method  in  mathematics, 
such  as  I  have  here  tried  to  realize,  and  which  could  well  be 
called  after  Euclid,  I  might  describe  thus.  It  cannot  indeed 
be  required  that  everything  should  be  proved,  since  that  is 
impossible;  but  one  can  see  to  it  that  all  propositions  which 
are  used  without  being  proved,  are  expressly  stated  as  such, 



so  that  it  is  clearly  known  on  what  the  whole  structure  rests. 
The  effort,  then,  must  be  to  reduce  the  number  of  these 
primitive  laws  as  far  as  possible,  by  proving  everything  which 
can  be  proved.  Further,  and  here  I  go  beyond  Euclid,  I 
require  that  all  methods  of  inference  and  deduction  which  are 
to  be  applied,  shall  be  previously  presented.  Otherwise  it  is 
impossible  to  ensure  with  certainty  that  the  first  requirement 
is  fulfilled.  I  think  that  I  have  now  attained  this  ideal  in 
essentials.  Stricter  requirements  could  only  be  made  in  a  few 
points.  In  order  to  secure  greater  mobility,  and  not  to  fall  into 
excessive  prolixity,  I  have  allowed  myself  to  make  tacit  use  of 
the  commutability  of  antecedents,  and  of  the  identification  of 
like  antecedents,  and  have  not  reduced  the  ways  of  inference 
and  deduction  to  the  smallest  number.  Those  who  know  my 
small  book  Begriffsschrift  will  be  able  to  gather  from  it  how 
the  strictest  requirements  could  be  forthcoming  here  as  well, 
but  also  that  this  would  bring  with  it  a  notable  increase  in 

Frege  is  correct  here  in  claiming  Euclid  as  his  predecessor,  insofar 
as  Euclid  was  the  first  to  carry  out  the  idea  of  an  axiomatic  system  in 
mathematics.  But  it  would  have  been  much  better  to  refer  to  Aristotle 
(14.02,  14.05),  for  what  Frege  offers  is  an  important  sharpening  of  the 
Aristotelian  concept  of  an  axiomatic  system.  His  first  requirement  is 
that  all  presuppositions  should  be  formulated  expressly  and  without 
gaps.  Then  he  makes  an  explicit  distinction  between  the  laws  and  the 
methods  of  inference  and  deduction,  i.e.  the  rules  of  inference.  This 
is  not  altogether  new  (cf.  22.12-22.15,  30.11.  §  31,  C),  but  is  stated 
with  greater  clarity  than  ever  before.  Finally.  Frege  can  be  con- 
trasted with  Leibniz,  Boole  and  other  earlier  writers  in  his  laying 
down  of  a  quite  new  requirement:  'considerably  higher  demands 
must  be  made  on  the  notion  of  proof  than  is  usual  in  arithmetic'. 
With  that,  mathematical  logic  enters  on  its  second  phase. 

38.23  and  38.24,  along  with  the  citations  from  Boole,  are  texts 
of  far-reaching  influence  on  the  concept  of  mathematical  logic.  In 
this  connection  two  further  quotations,  dating  from  1896,  may  be 

38.25  Words  such  as  'therefore',  'consequently',  'since' 
suggest  indeed  that  inference  has  been  made,  but  say  nothing 
of  the  principle  in  accordance  with  which  it  has  been  made,  and 
could  also  be  used  without  misuse  of  words  where  there  is  no 
logically  justified  inference.  In  an  inquiry  which  I  here  have 
in  view,  the  question  is  not  only  whether  one  is  convinced  of 
the  truth  of  the  conclusion,  with  which  one  is  usually  satisfied 



in  mathematics ;  but  one  must  also  bring  to  consciousness  the 
reason  for  this  conviction  and  the  primitive  laws  on  which  it 
rests.  Fixed  lines  on  which  the  deductions  must  move  are 
necessary  for  this,  and  such  are  not  provided  in  ordinary 

38.26  Inference  is  conducted  in  my  symbolic  system 
(Begriffsschrift)  according  to  a  kind  of  calculation.  I  do  not 
mean  this  in  the  narrow  sense,  as  though  an  algorithm  was  in 
control,  the  same  as  or  similar  to  that  of  ordinary  addition 
and  multiplication,  but  in  the  sense  that  the  whole  is  algorith- 
mic, with  a  complex  of  rules  which  so  regulate  the  passage 
from  one  proposition  or  from  two  such  to  another,  that  nothing 
takes  place  but  what  is  in  accordance  with  these  rules.  My 
aim,  therefore,  is  directed  to  continuous  strictness  of  proof  and 
utmost  logical  accuracy,  along  with  perspicuity  and  brevity. 

Frege's  program  of  thorough  proof  was  later  carried  out  in  mathe- 
matics by  Hilbert  with  a  view  to  pure  formality.  The  texts  can  be 
referred  to  in  0.  Becker  (38.27).  It  was  Lukasiewicz  who  applied 
it  to  logical  systems  with  complete  strictness.  We  shall  give  an 
example  in  the  chapter  on  propositional  logic  (43.45). 


The  idea  of  a  metalogic  was  an  inevitable  result  of  the  combination 
of  Boole's  formalism  and  Frege's  theory  of  proof.  For  once  formulae 
had  been  distinguished  from  rules,  and  the  former  treated  with  strict 
formalism,  'after  the  fashion  of  an  algorithm'  as  Frege  says,  then  the 
rules  had  to  be  interpreted  as  meaningful  and  having  content.  At 
once  the  rules  are  seen  as  belonging  to  a  different  level  to  the  for- 
mulae. The  notion  of  this  second  level  appears  first  in  connection 
with  mathematics  as  that  of  metamathematics  in  Hilbert.  We  cite  his 
lecture  Die  logischen  Grundlagen  der  Malhematik  (1923): 

38.28  The  basic  idea  of  my  theory  of  proof  is  this: 
Everything   that   goes   to    make    up   mathematics   in   the 
accepted  sense  is  strictly  formalized,   so  that  mathematics 
proper,   or  mathematics   in   the   narrower  sense,   becomes    a 
stock  of  formulae.  .  .  . 

Beyond  mathematics  proper,  formalized  in  this  way,  there 
is,  so  to  speak,  a  new  mathematics,  a  metamathematics,  which 
is  needed  to  establish  the  other  securely.  In  it,  by  contrast  to 
the  purely  formal  ways  of  inference  in  mathematics  proper, 



inference  which  has  regard  to  the  subject  matter  is  applied, 
though  merely  to  establish  the  freedom  from  contradiction  of 
the  axioms.  In  this  metamathematics  we  operate  with  the 
proofs  of  mathematics  proper,  these  last  themselves  forming 
the  object  of  the  inference  that  regards  the  matter.  In  this 
way  the  development  of  the  total  science  of  mathematics  is 
achieved  by  a  continual  exchange  which  is  of  two  kinds:  the 
gaining  of  new  provable  formulae  from  the  axioms  by  means  of 
formal  inference,  and  on  the  other  hand  the  addition  of  new 
axioms  along  with  the  proof  of  freedom  from  contradiction  by 
means  of  inference  having  regard  to  the  matter. 

The  axioms  and  provable  propositions,  i.e.  formulae,  which 
arise  in  this  process  of  exchange,  are  representations  of  the 
thoughts  which  constitute  the  usual  processes  of  mathematics 
as  understood  up  to  now,  but  they  are  not  themselves  truths 
in  an  absolute  sense.  It  is  the  insights  which  are  afforded  by  my 
theory  of  proof  in  regard  to  provability  and  freedom  from 
contradiction  which  are  rather  to  be  viewed  as  the  absolute 

This  important  text  goes  beyond  the  bounds  of  this  chapter  in 
that  it  touches  not  only  on  proof-theory  but  also  on  the  concept  of 
logic  and  its  relations  to  mathematics,  since  Hilbert  here  limits 
meaningful  inference  to  the  proof  of  freedom  from  contradiction, 
in  accordance  with  his  special  philosophy  of  mathematics.  The 
important  point  for  our  purpose  is  chiefly  the  sharp  distinction 
between  the  formalized,  and  so  in  Boole's  sense  meaningless, 
calculus  on  the  one  hand,  and  the  meaningful  rules  of  inference  on 
the  other.  This  idea,  too,  was  first  expressed  by  Frege,  when  he  requir- 
ed enumeration  of  all  'ways  of  inference  and  deduction'  as  distinct 
from  axioms  (cf.  38.24).  But  Frege  did  not  think  of  the  axioms  and 
theorems  as  meaningless,  however  formally  he  considered  them.  Here 
on  the  contrary  it  is  a  case  of  inscriptions  considered  purely  mate- 

A  new  stage  in  the  understanding  of  formalization  has  thus  been 
reached.  The  doctrine  in  Hilbert  is,  of  course,  limited  to  mathematics  - 
he  speaks  of  metamathematics.  But  soon  this  idea  was  to  be  extended 
to  logic,  and  this  came  about  in  the  Warsaw  School.  The  expression 
'metalogic'  first  occurs  in  a  paper  by  Lukasiewicz  and  Tarski  of 
1930  (38.29). 

Parallel  to  the  work  of  the  Warsaw  School  is  that  which  R.  Carnap 
was  carrying  on  in  Vienna  at  the  same  time. 

We  cite  now  a  text  of  Tarski's,  the  founder  of  systematic  meta- 
logic. He  forms  the  starting-point  for  the  most  recent  developments 
which  will  not  be  pursued  here. 



Tarski  wrote  in  1930: 

38.30  Our  object  in  this  communication  is  to  define  the 
meaning,  and  to  establish  the  elementary  properties,  of  some 
important  concepts  belonging  to  the  methodology  of  the  deductive 
sciences,  which,  following  Hilbert,  it  is  customary  to  call 
metamathematics . 

Formalized  deductive  disciplines  form  the  field  of  research  of 
metamathematics  roughly  in  the  same  sense  in  which  spatial 
entities  form  the  field  of  research  in  geometry.  These  discip- 
lines are  regarded,  from  the  standpoint  of  metamathematics, 
as  sets  of  sentences.  Those  sentences  which  (following  a  sug- 
gestion of  S.  Lesniewski)  are  also  called  meaningful  sentences, 
are  themselves  regarded  as  certain  inscriptions  of  a  well- 
defined  form.  The  set  of  all  sentences  is  here  denoted  by  the 
symbol  iSi.  From  the  sentences  of  any  set  X  certain  other 
sentences  can  be  obtained  by  means  of  certain  operations 
called  rules  of  inference.  These  sentences  are  called  the  conse- 
quences of  the  set  X.  The  set  of  all  consequences  is  denoted  by 
the  symbol  lCn  {X)\ 

An  exact  definition  of  the  two  concepts,  of  sentence  and  of 
consequence,  can  be  given  only  in  those  branches  of  meta- 
mathematics in  which  the  field  of  investigation  is  a  concrete 
formalized  discipline.  On  account  of  the  generality  of  the 
present  considerations,  however,  these  concepts  will  here  be 
regarded  as  primitive  and  will  be  characterized  by  means  of 
a  series  of  axioms. 


As  has  been  seen  above  (§  38)  Boole  (38.17),  Peirce  (38.19)  and 
with  them  the  other  mathematical  logicians  of  the  19th  century 
considered  logic  to  be  a  branch  of  mathematics,  this  last  being 
described  not  with  reference  to  its  object  but  its  method,  the 
application  of  a  calculus.  However,  at  the  end  of  the  19th  century 
there  arose  considerable  disagreement  about  the  relationship  of 
logic  to  mathematics,  a  disagreement  which  at  the  same  time 
concerned  the  answer  to  the  question  whether  logic  can  be  deve- 
loped purely  formally  as  a  system  of  symbols,  or  whether  it  necessa- 
rily involves  an  interpretation  of  the  symbols.  So  there  were  two 
problems,  but  both  concerned  with  the  concept  of  logic.  Three  main 
positions  took  shape:  the  logistic,  the  formalistic  (not  in  the  sense 
in  which  'formalism'  is  used  in  the  last  and  in  subsequent  sections) 



and  the  intuitionistic.  We  shall  illustrate  their  main  features  with 
some  texts. 


On  the  logistic  position  there  is  no  essential  distinction  between 
logic  and  mathematics,  inasmuch  as  mathematics  can  be  developed 
out  of  logic;  more  exactly,  inasmuch  as  all  mathematical  terms 
can  be  defined  by  logical  ones,  and  all  mathematical  theorems 
can  be  deduced  from  true  logical  axioms.  Frege  is  the  originator  of 
this  line  of  thought,  which  attained  its  fullest  development  in  the 
Principia  Malhematica  of  Whitehead  and  Russell,  written  precisely 
to  provide  a  thorough  proof  of  the  logistic  thesis. 

1.  Frege:  semantics 

Frege's  theory  of  logic  is  closely  connected  with  his  semantics 
(a  word  which  we  always  use  here  in  the  sense  of  Morris  (5.01),  not 
in  Tarski's  technical  sense).  On  this  point  we  shall  here  recall 
briefly  only  that  logic  for  Frege  was  not  a  game  with  symbols  but  a 
science  of  objective  thoughts  (Gedanken),  i.e.  of  ideal  propositions 
(and  so  of  lecta  in  the  sense  of  19.04ff.).  The  premisses  must  be  true, 
formalism  is  only  a  means.  To  begin  with,  we  give  a  text  about  the 
first  point: 

39.01  By  the  word  'sentence'  (Satz)  I  mean  a  sign  which  is 
normally  composite,  regardless  of  whether  the  parts  are 
spoken  words  or  written  signs.  This  sign  must  naturally  have 
a  sense  (Sinn).  I  shall  here  only  consider  sentences  in  which  we 
assert  or  state  something.  We  can  translate  a  sentence  into 
another  language.  In  the  other  language  the  sentence  is 
different  from  the  original  one,  since  it  consists  of  different 
components  (words)  differently  compounded;  but  if  the 
translation  is  correct,  it  expresses  the  same  sense.  And  the 
sense  is  properly  just  that  which  matters  to  us.  The  sentence 
has  a  value  for  us  through  the  sense  which  we  apprehend  in 
it,  and  which  we  recognize  as  the  same  in  the  translation  too. 
This  sense  I  call  'thought'  (Gedanke).  What  wre  prove  is  not 
the  sentence  but  the  thought.  And  it  makes  no  difference  what 
language  we  use  for  that  purpose.  In  mathematics  people 
speak  indeed  of  a  proof  of  a  Lehrsaiz  when  they  understand 
by  the  word  Satz  what  I  call  'thought'  -  or  perhaps  they  do  not 
sufficiently  distinguish  between  the  verbal  or  symbolic 
expression  and  the  thought  expressed.  But  for  clarity  it  is 
better  to  make  this  distinction.  The  thought  is  not  perceptible 
to  the  senses,  but  we  give  it  an  audible  or  visible  represen- 



tative  in  the  sentence.  Hence  I  say  'theorem'  rather  than 
'sentence',  'axiom'  rather  than  'primitive  sentence',  and  by 
theorems  and  axioms  I  understand  true  thoughts.  This  further 
implies  that  thoughts  are  not  something  subjective,  the 
product  of  our  mental  activity;  for  the  thought,  such  as  we 
have  in  the  theorem  of  Pythagoras,  is  the  same  for  everyone, 
and  its  truth  is  quite  independent  of  whether  it  is  or  is  not 
thought  by  this  or  that  man.  Thinking  is  to  be  viewed  not  as 
the  production  of  thought  but  as  its  understanding. 

Here,  in  another  terminology,  we  have  exactly  the  Stoic  doctrine 
that  logic  deals  with  lecta,  and  the  third  scholastic  view  (28.17) 
according  to  which  propositions  stand  for  ideal  structures. 

On  the  question  of  the  truth  of  premisses,  Frege  says : 

39.02  Nothing  at  all  can  be  deduced  from  false  premisses. 
A  mere  thought  which  is  not  accepted  as  true,  cannot  be  a 
premiss.  Only  when  I  have  accepted  a  thought  as  true  can  it 
be  a  premiss  for  me;  mere  hypotheses  cannot  be  used  as 
premisses.  Of  course,  I  can  ask  what  consequences  follow  from 
the  supposition  that  A  is  true  without  having  accepted  the 
truth  of  A ;  but  the  result  then  involves  the  condition :  if 
A  is  true.  But  that  is  only  to  say  that  A  is  not  a  premiss,  since 
a  true  premiss  does  not  occur  in  the  judgment  inferred. 

Frege  thus  holds  a  kind  of  absolutest  doctrine  closely  approximat- 
ing to  the  Aristotelian  theory  of  a7c68ei£i<;  (14.02)  but  apparently 
still  more  radical. 

We  append  now  a  characteristic  text  about  the  use  of  quotation- 
marks,  in  which  Frege's  high  degree  of  exactness  finds  expression  -  a 
degree  that  has  been  too  seldom  attained  since. 

39.03  People  may  perhaps  wonder  about  the  frequent  use 
of  quotation-marks ;  I  use  them  to  distinguish  the  cases  where 
I  am  speaking  of  the  symbol  itself,  from  those  where  I  am 
speaking  of  what  it  stands  for.  This  may  seem  very  pedantic, 
but  I  consider  it  necessary.  It  is  extraordinary  how  an  inexact 
manner  of  speaking  and  writing,  which  was  originally  perhaps 
used  only  for  convenience,  can  in  the  end  lead  thought 
astray  after  one  has  ceased  to  notice  it.  Thus  it  has  come  about 
that  numerals  are  taken  for  numbers,  names  for  what  they 
name,  what  is  merely  auxiliary  for  the  proper  object  of 
arithmetic.  Such  experiences  teach  us  how  necessary  it  is  to 
demand  exactness  in  ways  of  talking  and  writing. 


2.  Frege:  Logic  and  Mathematics 

39.04  Under  the  name  'formal  theory'  I  shall  here  consider 
two  modes  of  conception,  of  which  I  subscribe  to  the  first  and 
endeavour  to  refute  the  second.  The  first  says  that  all  arith- 
metical propositions  can,  and  hence  should,  be  deduced  from 
definitions  alone  by  purely  logical  means.  .  .  .  Out  of  all  the 
reasons  which  support  this  view  I  shall  here  adduce  only  one, 
which  is  based  on  the  comprehensive  applicability  of  arith- 
metical doctrines.  One  can  in  fact  number  pretty  well  every- 
thing that  can  be  an  object  of  thought:  the  ideal  as  well  as  the 
real,  concepts  and  things,  the  temporal  and  the  spatial,  events 
and  bodies,  methods  as  well  as  propositions;  numbers  them- 
selves can  be  in  turn  numbered.  Nothing  is  actually  required 
beyond  a  certain  definiteness  of  delimitation,  a  certain  logical 
completeness.  From  this  there  may  be  gathered  no  less  than 
that  the  primitive  propositions  on  which  arithmetic  is  based, 
are  not  to  be  drawn  from  a  narrow  domain  to  the  special 
character  of  which  they  give  expression,  as  the  axioms  of 
geometry  express  the  special  character  of  the  spatial  domain; 
rather  must  those  primitive  propositions  extend  to  everything 
thinkable,  and  a  proposition  of  this  most  universal  kind  is 
rightly  to  be  ascribed  to  logic. 

From  this  logical  or  formal  character  of  arithmetic  I  draw 
some  conclusions. 

First:  no  sharp  boundary  between  logic  and  arithmetic 
is  to  be  drawn;  considered  from  a  scientific  point  of  view 
both  constitute  a  single  science.  If  the  most  universal  primi- 
tive propositions  and  perhaps  their  immediate  consequences 
are  attributed  to  logic,  and  the  further  development  to 
arithmetic,  it  is  like  wanting  to  detach  a  special  science  of 
axioms  from  geometry.  Yet  the  partitioning  of  the  whole 
domain  of  knowledge  among  the  sciences  is  determined  not 
only  by  theoretical  but  also  by  practical  considerations,  so 
that  I  do  not  wish  to  say  anything  against  a  certain  practical 
separation.  But  it  must  not  become  a  breach  as  is  now  the 
case  to  the  detriment  of  both.  If  this  formal  theory  is  correct, 
logic  cannot  be  so  fruitless  as  it  may  appear  to  a  superficial 
consideration  -  of  which  logicians  are  not  guiltless.  And  there 
is  no  need  for  that  attitude  of  reserve  on  the  part  of  many 
mathematicians  towards  any  philosophic  justification  of 
whatever  is  real,  at  least  insofar  as  it  extends  to  logic.  This 



science  is  capable  of  no  less  exactness  than  mathematics 
itself.  On  the  other  hand  logicians  may  be  reminded  that  they 
cannot  learn  to  know  their  own  science  thoroughly  if  they 
do  not  trouble  themselves  about  arithmetic. 

39.05  My  second  conclusion  is  that  there  is  no  special 
arithmetical  kind  of  inference  such  that  it  cannot  be  reduced 
to  the  common  inference  of  logic. 

39.06  My  third  conclusion  concerns  definitions,  as  my 
second  concerned  kinds  of  inference.  In  every  definition 
something  has  to  presupposed  as  known,  by  means  of  which 
one  explains  what  is  to  be  understood  by  a  name  or  symbol. 
An  angle  cannot  be  well  defined  without  presupposing  know- 
ledge of  a  straight  line.  Now  that  on  which  a  definition  is 
based  may  itself  be  defined;  but  in  the  last  resort  one  must 
always  come  to  something  indefinable,  which  has  to  be 
recognized  as  simple  and  incapable  of  further  resolution.  And 
the  properties  which  belong  to  these  foundation  stones  of 
science,  contain  its  whole  content  in  embryo.  In  geometry 
these  properties  are  expressed  in  the  axioms,  to  the  extent 
that  these  are  independent  of  one  another.  Now  it  is  clear 
that  the  boundaries  of  a  science  are  determined  by  the 
nature  of  its  foundation  stones.  If,  as  in  geometry,  we  are 
originally  concerned  with  spatial  structures,  the  science,  too, 
will  be  limited  to  what  is  spatial.  Since  then  arithmetic  is  to 
be  independent  of  all  particular  properties  of  things,  that 
must  hold  for  its  foundations :  they  must  be  of  a  purely  logical 
kind.  The  conclusion  follows  that  everything  arithmetical 
is  to  be  reduced  by  definitions  to  what  is  logical. 

3.  Russell 

Frege's  postulates  were  first  taken  up  by  Giuseppe  Peano  -  though 
without  direct  dependence  on  Frege  -  then  by  Bertrand  Russell. 
The  latter  extended  the  logistic  thesis  to  geometry  and  mathematical 
disciplines  in  general. 

39.07  The  general  doctrine  that  all  mathematics  is  deduc- 
tion by  logical  principles  from  logical  principles  was  strongly 
advocated  by  Leibniz.  .  .  .  But  owing  partly  to  a  faulty  logic, 
partly  to  belief  in  the  logical  necessity  of  Euclidean  Geometry, 
he  was  led  into  hopeless  errors.  .  .  .  The  actual  propositions 
of  Euclid,  for  example,  do  not  follow  from  the  principles  of 
logic  alone;  ....  But  since  the  growth  of  non-Euclidean 
Geometry,  it  has  appeared  that  pure  mathematics  has  no 



concern  with  the  question  whether  the  axioms  and  propositions 
of  Euclid  hold  of  actual  space  or  not:  this  is  a  question  for 

applied  mathematics,  to  be  decided,  so  far  as  any  decision  is 
possible,  by  experiment  and  observation.  What  pure  mathe- 
matics asserts  is  merely  that  the  Euclidean  propositions  follow 
from  the  Euclidean  axioms  -  i.e.  it  asserts  an  implication: 
any  space  which  has  such  and  such  properties  has  also  such 
and  such  other  properties.  Thus,  as  dealt  with  in  pure  mathe- 
matics, the  Euclidean  and  non-Euclidean  Geometries  are 
equally  true:  in  each  nothing  is  affirmed  except  implica- 
tions. .  .  . 

39.08  Thus  pure  mathematics  must  contain  no  indefinables 
except  logical  constants,  and  consequently  no  premisses,  or 
indemonstrable  propositions,  but  such  as  are  concerned 
exclusively  with  logical  constants  and  with  variables.  It  is 
precisely  this  that  distinguishes  pure  from  applied  mathe- 

How  and  to  what  extent  this  program  was  carried  out,  cannot 
here  be  pursued.  Reference  may  be  made  to  Becker  (39.09).  In 
conclusiou  we  should  like  only  to  illustrate  Frege's  definition  of 
number  by  means  of  purely  logical  concepts,  especially  with  a  view 
to  comparing  it  with  a  similar  discovery  in  the  Indian  logic  of  the 
17th  century  (54.17). 

4.  Frege:  number 

39.10  To  illuminate  matters  it  will  be  good  to  consider 
number  in  connection  with  a  judgment  where  its  primitive 
manner  of  application  occurs.  If  when  I  see  the  same  outward 
appearances  I  can  say  with  the  same  truth:  'this  is  a  group  of 
trees'  and  'these  are  five  trees'  or  'here  are  four  companies' 
and  'here  are  500  men',  no  difference  is  made  to  the  individual 
or  to  the  whole,  the  aggregate,  but  to  my  naming.  But  this 
is  only  the  sign  of  the  substitution  of  one  concept  by  another. 
This  suggests  an  answer  to  the  first  question  of  the  previous 
paragraph,  that  number  involves  a  statement  about  a  concept. 
This  is  perhaps  most  evident  for  the  number  0.  When  I  say: 
'Venus  has  0  moons',  there  is  no  moon  or  aggregate  of  moons 
there  about  which  anything  can  be  said;  but  to  the  concept 
'moon  of  Venus'  there  is  attributed  a  property,  viz.  that  of 
comprising  nothing  under  it.  When  I  say:  'the  emperor's 
carriage  is  drawn  by  four  horses',  I  apply  the  number  four  to 
the  concept  'horse  which  draws  the  emperor's  carriage'.  .  .  . 



39.11  Among  the  properties  which  are  predicated  of  a 
concept  I  do  not,  of  course,  understand  the  notes  which  make 
up  the  concept.  These  are  properties  of  the  things  which  fall 
under  the  concept,  not  of  the  concept.  Thus  'right-angled' 
is  not  a  property  of  the  concept  'right-angled  triangle';  but 
the  proposition  that  there  is  no  right-angled,  rectilineal, 
equilateral  triangle,  states  a  property  of  the  concept  'right- 
angled,  rectilineal,  equilateral  triangle',  attributing  to  it  the 
number  0. 

39.12  In  this  respect  existence  is  like  number.  The  affir- 
mation of  existence  is  nothing  else  than  the  denial  of  the 
number  0.  Since  existence  is  a  property  of  the  concept,  the 
ontological  proof  of  the  existence  of  God  fails  of  its  purpose 

It  would  also  be  false  to  deny  that  existence  and  unicity 
can  ever  be  notes  of  concepts.  They  are  only  not  notes  of  that 
concept  to  which  the  manner  of  speech  might  lead  one  to 
ascribe  them.  E.g.  when  all  concepts  belonging  only  to  one 
object  are  collected  under  one  concept,  uniqueness  is  a  note  of 
this  concept.  The  concept  'moon  of  the  earth',  for  instance, 
would  fall  under  it,  but  not  the  so-called  heavenly  body. 
Thus  a  concept  can  be  allowed  to  fall  under  a  higher  one, 
under  a  concept,  so  to  speak,  of  second  order.  But  this  rela- 
tionship is  not  to  be  confused  with  that  of  subordination. 

Frege's  definition  of  number  was  later  interpreted  by  Russell 
extensionally,  when  he  took  numbers  as  classes  of  classes  (39.13). 


The  formalists,  too,  see  no  essential  difference  between  logical 
and  mathematical  formulae,  but  they  understand  both  forma- 
listically  and  think  of  the  single  system  composed  of  them  as  a 
system  of  symbols.  Evidence  and  truth  of  the  axioms  have  no  part 
to  play:  but  freedom  from  contradiction  is  everything.  The  founder 
of  formalism  is  David  Hilbert,  the  essentials  of  whose  thought  on 
logic  is  contained  in  the  text  given  earlier  (38.28).  Here  we  add  only 
a  brief  passage  from  a  letter  to  Frege  in  1899  or  1900: 

39.14  You  write:  'From  the  truth  of  the  axioms  it  follows 
that  they  do  not  contradict  one  another'.  I  was  very  interested 
to  read  this  particular  sentence  of  yours,  because  for  my  part, 
ever  since  I  have  been  thinking,  writing  and  lecturing  about 
such  matters,  I  have  been  accustomed  to  say  just  the  reverse: 
if  the  arbitrarily  posited  axioms  are  not  in  mutual  (sic)  con- 



tradiction  with  the  totality  of  their  consequences,  then  they 
are  true  -  the  things  defined  by  the  axioms  exist.  That  for 
me  is  the  criterion  of  truth  and  existence. 

For  the  rest,  it  is  not  easy  to  find  texts  to  illustrate  Hilbert's 
thought  before  1930;  for  that  and  the  later  development  Becker 
may  again  be  consulted  (39.15). 

It  should  be  noted  that  formalism  has  been  very  important  for  the 
concept  of  logic,  quite  apart  from  its  value  as  a  theory.  Logic  having 
been  previously  viewed  as  a  calculus,  it  is  henceforth  ever  more  and 
more  transposed  onto  the  level  of  metalogic.  After  Hilbert,  it  is  not 
the  formulae  themselves  but  the  rules  of  operation  by  which  they 
are  formed  and  derived  that  are  more  and  more  made  the  object  of 
logical  investigation. 


By  contrast  to  the  logisticians  and  formalists,  the  intuition ists 
make  a  sharp  distinction  between  logic  and  mathematics.  Mathe- 
matics is  not,  for  them,  a  set  of  formulae,  but  primarily  a  mental 
activity  the  results  of  which  are  subsequently  communicable  by 
means  of  language.  In  language,  as  used  by  mathematicians, 
certain  regularities  are  observed,  and  this  leads  to  the  development 
of  a  logic.  Thus  logic  is  not  presupposed  by,  but  abstracted  from 
mathematics.  Once  that  has  been  done,  it  can  then  be  formalized, 
but  this  is  a  matter  of  secondary  importance.  * 

Intuitionism  has  a  fairly  long  history  in  mathematics:  L.  Kron- 
ecker  and  H.  Poincare  are  precursors;  H.  Weyl  is  reckoned  a 
'semi-intuitionist'.  But  L.  E.  J.  Brouwer  ranks  as  the  founder  of 
the  school,  and  intuitionistic  logic  was  first  properly  formulated 
(and  formalized)  by  A.  Heyting  in  1930. 

From  the  standpoint  of  formal  logic  it  is  to  be  noted  that  the 
intuitionists,  as  they  themselves  say,  admit  the  principle  of  tertium 
exclusion  only  under  certain  limitations.  In  this  respect  their 
doctrine  belongs  to  those  'heterodox'  logics  of  which  we  shall  speak 
in  §  49. 

We  give  one  text  from  Heyting  and  one  from  Brouwer : 

39.16  Intuitionistic  mathematics  is  an  activity  of  thought. 
and  every  language  -  even  the  formalistic  -  is  for  it  only  a 
means  of  communication.  It  is  impossible  in  principle  to 
establish  a  system  of  formulae  that  would  have  the  same 
value  as  intuitionistic  mathematics,  since  it  is  impossible  to 

*  Special  thanks  are  due  to  Prof.  E.  W.  Beth  for  much  information  in  this 
connection,  as  generally  for  his  help  with  the  composition  of  this  fifth  part. 



reduce  the  possibilities  of  thought  to  a  finite  number  of 
rules  that  thought  can  previously  lay  down.  The  endeavour 
to  reproduce  the  most  important  parts  of  mathematics  in  a 
language  of  formulae  is  justified  exclusively  by  the  great 
conciseness  and  defmiteness  of  this  last  as  compared  with 
customary  languages,  properties  which  fit  it  to  facilitate 
penetration  of  the  intuitionistic  concepts  and  their  applica- 
tion in  research. 

For  constructing  mathematics  the  statement  of  universally 
valid  logical  laws  is  not  necessary.  These  laws  are  found  as  it 
were  anew  in  every  individual  case  to  be  valid  for  the  mathe- 
matical system  under  consideration.  But  linguistic  com- 
munication moulded  according  to  the  needs  of  everyday  life 
proceeds  according  to  the  form  of  logical  laws  which  it 
presupposes  as  given.  A  language  which  imitated  the  process 
of  intuitionistic  mathematics  step  by  step  would  so  diverge  in 
all  its  parts  from  the  usual  pattern  that  it  would  have  to 
surrender  again  all  the  useful  properties  mentioned  above. 
These  considerations  have  led  me  to  begin  the  formalization 
of  intuitionistic  mathematics  once  again  with  a  propositional 

The  formulae  of  the  formalistic  systems  come  into  being  by 
the  application  of  a  finite  number  of  rules  of  operation  to  a 
finite  number  of  axioms.  Besides  'constant'  symbols  they  also 
contain  variables.  The  relationship  between  this  system  and 
mathematics  is  this,  that  on  a  determinate  interpretation 
of  the  constants  and  under  certain  restrictions  on  substitu- 
tion for  variables  every  formula  expresses  a  correct  mathe- 
matical proposition.  (E.g.  in  the  propositional  calculus  the 
variables  must  be  replaced  only  by  senseful  mathematical 
sentences.)  If  the  system  is  so  constructed  as  to  fulfil  the 
last-mentioned  requirement,  its  freedom  from  contradiction 
is  thereby  guaranteed,  in  the  sense  that  it  cannot  contain 
any  formula  which  would  express  a  contradictory  proposition 
on  that  interpretation. 

The  formalistic  system  can  also  be  considered  mathemati- 
cally for  its  own  sake,  without  reference  to  any  interpretation. 
Freedom  from  contradiction  then  takes  on  a  new  meaning 
inasmuch  as  contradiction  is  defined  as  a  definite  formula; 
for  us  this  method  of  treatment  is  less  to  the  fore  than  the 
other.  But  here  questions  come  in  about  the  independence  and 
completeness  of  the  axiom-system. 



39.17  The  differences  about  the  Tightness  of  the  new 
formalistic  foundations  and  the  new  intuitionistic  construc- 
tion of  mathematics  will  be  removed,  and  the  choice  between 
the  two  methods  of  operation  reduced  to  a  matter  of  taste, 
as  soon  as  the  following  intuitions  (Einsichten)  have  been 
generally  grasped.  They  primarily  concern  formalism,  but 
were  first  formulated  in  intuitionist  literature.  This  grasp 
is  only  a  matter  of  time,  since  they  are  results  purely  of 
reflection,  containing  nothing  disputable,  and  necessarily 
acknowledged  by  everyone  who  has  once  understood  them. 
Of  the  four  intuitions  this  understanding  and  acknowledge- 
ment has  so  far  been  attained  for  two  in  the  formalistic 
literature.  Once  the  same  state  of  affairs  has  been  reached  for 
the  other  two,  an  end  will  have  been  put  to  disputes  about 
foundations  in  mathematics. 

FIRST  INTUITION.  The  distinction  between  the  formalistic 
endeavours  to  construct  the  'mathematical  slock  of  formulae' 
(formalistic  idea  of  mathematics)  and  an  intuitive  (meaningful) 
theory  of  the  laws  of  this  construction,  as  also  the  understanding 
that  for  the  last  theory  the  intuitionistic  mathematics  of  the  set 
of  natural  numbers  is  indispensable. 

SECOND  INTUITION.  The  rejection  of  the  thoughtless 
application  of  the  logical  theorem  of  tertium  exclusum,  as  also 
the  awareness  first,  that  investigation  of  the  credentials  and 
domain  of  validity  of  the  said  theorem  constitutes  an  essential 
object  of  mathematical  foundational  research;  second,  that  this 
domain  of  validity  in  intuitive  (meaningful)  mathematics  com- 
prises only  finite  systems. 

THIRD  INTUITION.  The  identification  of  the  theorem  of 
tertium  exclusum  with  the  principle  of  the  solubility  of  every 
mathematical  problem. 

FOURTH  INTUITION.  The  awareness  that  the  (meaningful) 
justification  of  formalistic  mathematics  through  proof  of  its 
freedom  from  contradiction  involves  a  vicious  circle,  since  this 
justification  depends  on  the  (meaningful)  correctness  of  the 
proposition  that  the  correctness  of  a  proposition  follows  from 
the  freedom  from  contradiction  of  this  proposition,  i.e.  from  the 
(meaningful)  correctness  of  the  theorem  of  tertium  exclusum. 




The  system  of  mathematical  logic  inaugurated  by  Boole  in  1847 
holds  a  special  place  in  history  in  that  it  admits  of  two  interpreta-     / 
tions,  in  class-logic  and  propositional  logic.  In  this  section  we  shall  ^ 
consider  the  abstract  calculus  itself  and  its  classical  interpretation, 
reserving  the  propositional  interpretation  to  the  following  section. 

The  growth  of  Boole's  calculus  can  be  summarized  as  follows: 
De  Morgan  is  its  precursor  (though  his  chief  work  was  published 
contemporaneously  with  Boole's  in  1847);  Boole  set  out  the  main 
lines  of  the  system  in  that  year;  but  his  exposition  lacks  the  concept 
of  the  logical  sum  which  first  appears  in  Peirce  (1867),  Schroder 
(1877),  and  Jevons  (1890),  as  also  the  concept  of  inclusion,  originally 
introduced  by  Gergonne  (1816)  and  clearly  formulated  by  Peirce  in 
1870.  Schroder's  system  (1890)  ranks  as  the  completion  of  this 
growth,  though  perhaps  Peano's  (1899)  may  here  be  counted  as  the 
real  close. 


Boole's  calculus  emerged  in  a  way  from  the  'classical'  endeavours 
to  broaden  the  Aristotelian  syllogistic  (36.15 f.).  This  can  be  most 
clearly  seen  from  the  syllogistic  of  Augustus  de  Morgan. 

40.01  I  shall  now  proceed  to  an  enlarged  view  of  the 
proposition,  and  to  the  structure  of  a  notation  proper  to  repre- 
sent its  different  cases. 

As  usual,  let  the  universal  affirmative  be  denoted  by  A, 
the  particular  affirmative  by  /,  the  universal  negative  by  E, 
and  the  particular  negative  by  O.  This  is  the  extent  of  the 
common  symbolic  expression  of  propositions:  I  propose  to 
make  the  following  additions  for  this  work.  Let  one  particular 
choice  of  order,  as  to  subject  and  predicate,  be  supposed 
established  as  a  standard  of  reference.  As  to  the  letters  X,  F,. 
Z,  let  the  order  always  be  that  of  the  alphabet,  IF,  YZ,  XZ 
Let  x,  y,  z,  be  the  contrary  names  of  X,  F,  Z;  and  let  the 
same  order  be  adopted  in  the  standard  of  reference.  Let  the 
four  forms  when  choice  is  made  of  an  X,  F,  Z,  be  denoted  by 
At,  E  ,  /,,  O,;  but  when  the  choice  is  made  from  the  contraries, 
let  them  be  denoted  by  A',  E\  /',  O' .  Thus  with  reference  to 
Y  and  Z,  "Every  Y  is  Z"  is  the  At  of  that  pair  and  order: 
while  "Every  y  is  z"  is  the  A'.  I  should  recommend  ^and^' 



to  be  called  the  sub-A  and  the  super- A  of  the  pair  and  order 
in  question:  the  helps  which  this  will  give  the  memory  will 
presently  be  very  apparent.  And  the  same  of  I t  and  /',  etc. 
Let  the  following  abbreviations  be  employed;  -  X)  Y 
means  "Every  X  is  Y".  X.  Y  means  "No  X  is  Y".  X:Y  means 
"Some  Xs  are  not  Ys".   XY  means  "Some  Xs  are  Fs". 

Later,  De  Morgan  developed  a  different  symbolism.  We  give  it- 
description  and  a  comparative  table,  from  a  paper  of  18.rjf): 

40.02  Let  the  subject  and  predicate,  when  specified,  be 
written  before  and  after  the  symbols  of  quantity.  Let  the 
enclosing  parenthesis,  as  in  X)  or  (X,  denote  that  the  name- 
symbol  X,  which  would  be  enclosed  if  the  oval  were  completed, 
enters  universally.  Let  an  excluding  parenthesis,  as  in  )X 
or  X(,  signify  that  the  name-symbol  enters  particularly.  Let 
an  even  number  of  dots,  or  none  at  all,  inserted  between 
the  parentheses,  denote  affirmation  or  agreement;  let  an  odd 
number,  usually  one,  denote  negation  or  non-agreement. 




Notation  of     Both. 




my  Work  on 


expressed  in 





A1                  X)Y 


Every  X  is  F 


A1                     x  )  y  or 

x  ))  y  or 

Every  F  is  X 


X((  Y 


E1                     X)yov 


No  X  is  F 

X  .  Y 

X).(  Y 


E1                     x  )  For 


Everything  is 


X(.)  Y 

X  or  F  o  rboth 



It                   XY 


Some  Xs  are  Fs 


I1                    xy 


Some    things 


are   neither  .Ys 
nor  Fs 


Ox                   Xy  or 


Some    Xs    are 

X  :  F 

X  (.(  Y 

not  Fs 


O1                   xY  or 

x  ()  F  or 

Some  Fs  are 

F  :  X 


not  Xs 



Boole,  who  was  the  first  to  outline  clearly  the  program  of  mathe- 
matical logic,  was  also  the  first  to  achieve  a  partial  execution.  In 
this  respect  there  is  a  great  likeness  between  his  relationship  to 
Leibniz  and  that  of  Aristotle  to  Plato.  For  with  Boole  as  with 
Aristotle  we  find  not  only  ideas  but  a  system. 

This  system  of  Boole's  can  be  described  thus:  it  is  in  the  first 
r^ace  closely  allied  to  arithmetic,  in  that  it  uses  only  arithmetical 
symbols  and  has  only  one  law  that  diverges  from  those  of  arith- 
metic, viz.  xn  =  x.  All  its  procedures  are  taken  over  from  simple 
algebra;  Boole  has  no  conscious  awareness  of  purely  logical  methods 
(even  of  those  which  are  intuitively  used  in  algebra),  e.g.  of  the  rules 
of  detachment  and  substitution.  As  a  matter  of  fact,  even  the  basic 
law  mentioned  makes  very  little  difference  to  the  algebraic  character 
of  his  system  -  which  is  algebra  limited  to  the  numbers  0  and  1. 

Boole's  mathematicism  goes  so  far  -  and  this  is  the  second  main 
characteristic  of  his  doctrine  -  that  he  introduces  symbols  and 
procedures  which  admit  of  no  logical  interpretation,  or  only  a  com- 
plicated and  scarcely  interesting  one.  Thus  we  meet  with  subtrac- 
tion and  division  and  numbers  greater  than  1. 

From  the  logical  point  of  view  it  is  to  be  noted  that  disjunction 
(symbolized  by  lx  +  y')  is  taken  as  exclusive,  and  that  inclusion  is 
expressed  by  means  of  equality.  Both  lead  to  difficulties  and 
unnecessary  complications;  both  are  the  result  of  the  tendency 
to  mathematicize. 

A  third  and  special  characteristic  is  that  the  system  possesses 
two  interpretations,  in  classical  and  propositional  logic. 

Altogether,  in  spite  of  its  defects,  Boolean  algebra  is  a  very 
successful  piece  of  logic.  Boole  resembles  Aristotle  both  in  point  of 
originality  and  fruitfulness ;  indeed  it  is  hard  to  name  another 
logician,  besides  Frege,  who  has  possessed  these  qualities  to  the 
same  degree,  after  the  founder. 

1.  Symbolism  and  basic  concepts 

40.04  Proposition  I.  All  the  operations  of  the  Language, 
as  an  instrument  of  reasoning,  may  be  conducted  by  a  system 
of  signs  composed  of  the  following  elements,  viz. : 

1st.  Literal  symbols  as  x,  y,  etc.,  representing  things  as 
subjects  of  our  conceptions. 

2nd.  Signs  of  operation,  as  +,  -,  x,  standing  for  those 
operations  of  the  mind  by  which  the  conceptions  of  things 
are  combined  or  resolved  so  as  to  form  new  conceptions 
involving  the  same  elements. 

3rd.  The  sign  of  identity,  =. 



And  these  symbols  of  Logic  are  in  their  use  subject  to 
definite  laws,  partly  agreeing  with  and  partly  differing  from 
the  laws  of  the  corresponding  symbols  in  the  science  of 

40.05  Let  us  employ  the  symbol  1  or  unity,  to  represent 
the  Universe,  and  let  us  understand  it  as  comprehending 
every  conceivable  class  of  objects  whether  actually  existing 
or  not,  it  being  premised  that  the  same  individual  may  La 
found  in  more  than  one  class,  inasmuch  as  it  may  possess 
more  than  one  quality  in  common  with  other  individuals. 
Let  us  employ  the  letters  X,  Y,  Z,  to  represent  the  individual 
members  of  classes,  X  applying  to  every  member  of  one 
class,  as  members  of  that  particular  class,  and  Y  to  every 
member  of  another  class  as  members  of  such  class,  and  so  on, 
according  to  the  received  language  of  treatises  on  Logic. 

Further  let  us  conceive  a  class  of  symbols  x,  y,  z,  possessed 
of  the  following  character. 

The  symbol  x  operating  upon  any  subject  comprehending 
individuals  or  classes,  shall  be  supposed  to  select  from  that 
subject  all  the  Xs  which  it  contains.  In  like  manner  the 
symbol  y,  operating  upon  any  subject,  shall  be  supposed  to 
select  from  it  all  individuals  of  the  class  Y  which  are  com- 
prised in  it,  and  so  on. 

When  no  subject  is  expressed,  we  shall  suppose  1  (the 
Universe)  to  be  the  subject  understood,  so  that  we  shall 

40.051  x  =  x  (1), 

the  meaning  of  either  term  being  the  selection  from  the 
Universe  of  all  the  Xs  which  it  contains,  and  the  result  of 
the  operation  being  in  common  language,  the  class  X,  i.e.  the 
class  of  which  each  member  is  an  X . 

From  these  premises  it  will  follow,  that  the  product  xy  will 
represent,  in  succession,  the  selection  of  the  class  Y,  and  the 
selection  from  the  class  Y  of  such  individuals  of  the  class  X 
as  are  contained  in  it,  the  result  being  the  class  whose  members 
are  both  Xs  and  Fs.  .  .  . 

From  the  nature  of  the  operation  which  the  symbols  x, 
y,  z,  are  conceived  to  represent,  we  shall  designate  them  as 
elective  symbols.  An  expression  in  which  they  are  involved 
will  be  called  an  elective  function,  and  an  equation  of  which 
the  members  are  elective  functions,  will  be  termed  an  elective 
equation.  .  .  . 



1st.  The  result  of  an  act  of  election  is  independent  of  the 
grouping  or  classification  of  the  subject. 

Thus  it  is  indifferent  whether  from  a  group  of  objects 
considered  as  a  whole,  we  select  the  class  X,  or  whether  we 
divide  the  group  into  two  parts,  select  the  Xs  from  them 
separately,  and  then  connect  the  results  in  one  aggregate 

We  may  express  this  law  mathematically  by  the  equation 

(40.052)  x  (u  +  v)  =  xu  +  xv, 

u  +  v  representing  the  undivided  subject,  and  u  and  v  the 
component  parts  of  it. 

2nd.  It  is  indifferent  in  what  order  two  successive  acts  of 
election  are  performed. 

Whether  from  the  class  of  animals  we  select  sheep,  and 
from  the  sheep  those  which  are  horned,  or  whether  from  the 
class  of  animals  we  select  the  horned,  and  from  these  such  as 
are  sheep,  the  result  is  unaffected.  In  either  case  we  arrive 
at  the  class  of  horned  sheep. 

The  symbolical  expression  of  this  law  is 

(40.053)  xy  =  yx. 

3rd.  The  result  of  a  given  act  of  election  performed  twice, 
or  any  number  of  times  in  succession,  is  the  result  of  the 
same  act  performed  once.  .  .  .  Thus  we  have 

(40.054)  xx  =  x, 
or     x2  =  x: 

and  supposing  the  same  operation  to  be  n  times  performed, 
we  have 

(40.055)  xn  =x, 

which  is  the  mathematical  expression  of  the  law  above 

The  laws  we  have  established  under  .  .  .  symbolical  forms  .  .  . 
are  sufficient  for  the  base  of  a  Calculus.  From  the  first  of 
these  it  appears  that  elective  symbols  are  distributive,  from 
the  second  that  they  are  commutative;  properties  which  they 
possess  in  common  with  symbols  of  quantity,  and  in  virtue 
of  which,  all  the  processes  of  common  algebra  are  applicable 
to  the  present  system.  The  one  and  sufficient  axiom  involved 
in  this  application  is  that  equivalent  operations  performed 
upon  equivalent  subjects  produce  equivalent  results. 

The  third  law  ...  we  shall  denominate  the  index  law.  It 
is  peculiar  to  elective  symbols. 


2.  Applications 

We  now  give  two  examples  of  the  application  of  these  principles 
in  Boole's  work.  The  first  concerns  the  law  of  contradiction. 

40.06  That  axiom  of  metaphysicians  which  is  termed  the 
principle  of  contradiction,  and  which  affirms  that  it  is  impos- 
sible for  any  being  to  possess  a  quality,  and  at  the  same  time 
not  to  possess  it,  is  a  consequence  of  the  fundamental  law 
of  thought,  whose  expression  is  x2  =  x. 

Let  us  write  this  equation  in  the  form 

(40.061)  x-x2  =  0 
whence  we  have 

(40.062)  x(l  -x)  =  0;  (1) 

both  these  transformations  being  justified  by  the  axiomatic 
laws  of  contradiction  and  transposition.  .  .  .  Let  us  for  simpli- 
city of  conception,  give  to  the  symbol  x  the  particular  inter- 
pretation of  men,  then  1  -  x  will  represent  the  class  of  'not- 
men'.  .  .  .  Now  the  formal  product  of  the  expressions  of  two 
classes  represents  that  class  of  individuals  which  is  common 
to  them  both.  .  .  .  Hence  x  (1  -  x)  will  represent  the  class 
whose  members  are  at  once  'men',  and  'not-men',  and  the 
equation  (1)  thus  expresses  the  principle,  that  a  class  whose 
members  are  at  the  same  time  men  and  not  men  does  not  exist. 
In  other  words,  that  it  is  impossible  for  the  same  individual  to 
be  at  the  same  time  a  man  and  not  a  man.  Now  let  the  meaning 
of  the  symbol  x  be  extended  from  the  representing  of  'men', 
to  that  of  any  class  of  beings  characterized  by  the  possession  of 
any  quality  whatever;  and  the  equation  (1)  will  then  express 
that  it  is  impossible  for  a  being  to  possess  a  quality  and  not 
to  possess  that  quality  at  the  same  time.  But  this  is  identically 
that  'principle  of  contradiction'  which  Aristotle  has  de- 
scribed as  the  fundamental  axiom  of  all  philosophy.  .  .  . 

The  above  interpretation  has  been  introduced  not  on 
account  of  its  immediate  value  in  the  present  system,  but  as 
an  illustration  of  a  significant  fact  in  the  philosophy  of  the 
intellectual  powers,  viz.,  that  what  has  been  commonly  re- 
garded as  the  fundamental  axiom  of  metaphysics  is  but  the 
consequence  of  a  law  of  thought,  mathematical  in  its  form. 

The  second  example  is  taken  from  the  application  of  Boole's 
methods  in  the  domain  of  syllogistic. 

40.07  The  equation  by  which  we  express  any  Proposition 
concerning  the  classes  X  and  Y,  is  an  equation  between  the 



symbols  x  and  y,  and  the  equation  by  which  we  express  any 
proposition  concerning  the  classes  Y  and  Z,  is  an  equation 
between  the  symbols  y  and  z.  If  from  two  such  equations  we 
eliminate  y,  the  result,  if  it  do  not  vanish,  will  be  an  equation 
between  x  and  z,  and  will  be  interpretable  into  a  Proposition 
concerning  the  classes  X  and  Z.  And  it  will  then  constitute 
the  third  member,  or  Conclusion,  of  a  Syllogism,  of  which 
the  two  given  Propositions  are  the  premises. 

The  result  of  the  elimination  of  y  from  the  equations 

ay  +  b  =  0, 


a'  y  -  b'  =  0, 
is  the  equation  ab'  -  a'  b  =  0     (15). 

40.08  Ex(ample).  AA,  Fig.  1,  and  by  mutation  of  premises 
(change  of  order),  AA,  Fig.  4. 

All  Ys  are  Xs,  y  (1  -  x)  =  0,  or  (1  -  x)  y  =  0, 
All  Zs  are  Ys,  z  (1  -  y)  =  0,  or  zy  -  z  =  0. 
Eliminating  y  by  (15)  we  have 
z  (1  -  x)  =  0, 
All  Zs  are  Xs. 

In  both  these  texts  Boolean  methods  are  being  applied  to  tradi- 
tional problems,  involving  logical  relationships  between  two  objects 
(classes,  propositions).  But  the  interesting  thing  about  this  calculus 
for  our  history  is  that  it  is  applicable  to  more  than  two  objects, 
so  that  it  oversteps  the  limits  of  the  'classical'  logic.  An  instance 
is  given  later  (41.03). 


The  original  Boolean  calculus  had  two  main  defects  from  the 
logical  point  of  view,  both  occasioned  by  its  extreme  mathemati- 
cism;  disjunction  was  treated  as  exclusive,  and  there  was  no  symbol 
to  hand  for  inclusion,  though  that  is  fundamental  in  logic.  The  first 
defect  was  remedied  by  Jevons,  who  was  strongly  opposed  to  this 
mathematicism  and  introduced  non-exclusive  disjunction. 

49.09  There  are  no  such  operations  as  addition  and  sub- 
traction in  pure  logic.  .  . 

40.10  Now  addition,  subtraction,  multiplication,  and  divi- 
sion, are  alike  true  as  modes  of  reasoning  in  numbers,  where 
we  have  the  logical  condition  of  a  unit  as  a  constant  restriction. 
But  addition  and  subtraction  do  not  exist,  and  do  not  give 
true  results  in  pure  logic,  free  from  the  conditions  of  number. 



For  instance  take  the  logical  proposition  - 
meaning  what  is  either  A  or  B  or  C  is  either  A  or  D  or  E,  and 
vice  versa.  There  being  no  exterior  restriction  of  meaning 
whatever,  except  that  some  terms  must  always  have  the  same 
meaning,  we  do  not  know  which  of  A,  D,  E,  is  B,  nor  which 
is  C;.  .  .  .  The  proposition  alone  gives  us  no  such  information. 

Much  clearer  is  Charles  S.  Peirce,  also  an  opponent  of  Boole's 
mathematicism  (1867).  He  uses  an  appropriate  though  still  primitive 

40.11  Let  the  sign  of  equality  with  a  comma  beneath  it 
express  numerical  identity.  .  .  .  Let  a  -t  b  denote  all  the 
individuals  contained  under  a  and  b  together.  The  operation 
here  performed  will  differ  from  arithmetical  addition  in  two 
respects :  first,  that  it  has  reference  to  identity,  not  to  equality, 
and  second,  that  what  is  common  to  a  and  b  is  not  taken  into 
account  twice  over,  as  it  would  be  in  arithmetic.  The  first  of 
these  differences,  however,  amounts  to  nothing,  inasmuch  as 
the  sign  of  identity  would  indicate  the  distinction  in  which  it 
is  founded;  and  therefore  we  may  say  that 

(1)  If  No  a  is  b       a  -t  6  ^  a  +  b. 
It  is  plain  that 

(2)  a  -t  a  f  fl 

and  also,  that  the  process  denoted  by  -t,  and  which  I  shall  call 
the  process  of  logical  addition,  is  both  commutative  and 
associative.  That  is  to  say 

(3)  a  -b  b  =?  b  -t  a 

(4)  (a  h?  b)  -t  c  7=  a  -t  (b  -t  c). 

This  is  the  third  time  that  non-exclusive  disjunction  is  discovered, 
cf.  Galen  (20.18)  and  Burleigh  (30.20). 

A  symbolism  quite  different  from  that  of  mathematics  is  first  met 
with  in  Peano  (41.20). 


The  introduction  of  the  concept  of  inclusion  and  a  symbol  for  it 
has  a  fairly  long  history.  The  modern  symbol  appears  thirty  years 
before  Boole's  Analysis  and  quite  independently  of  his  calculus 
in  J.  D.  Gergonne's  Essai  de  dialedique  ralionelle,  1816/17.  (The 
parentheses  enclosing  the  italic  capitals  in  this  text  are  Gergonne's.) 



40.12  We  have  chosen  the  signs  to  characterize  these  rela- 
tions in  the  way  which  seems  best  for  linking  the  sign  to  the 
thing  signified,  and  this  is  an  endeavour  which  we  think  of 
some  importance,  however  puerile  it  may  appear  at  first.  The 
letter  (H),  initial  letter  of  the  word  Hors  (outside)  designates 
the  system  of  two  ideas  completely  outside  one  another,  as  are 
the  two  vertical  strokes  of  this  letter.  These  two  strokes  can 
next  be  considered  as  crossed  to  form  the  letter  (X)  intended 
to  recall  the  system  of  two  ideas  which,  as  it  were,  somehow 
intersect.  Finally  the  two  strokes  can  be  identified  so  as  to 
form  the  letter  (/)  which  we  use  to  represent  the  system  of 
two  ideas  which  exactly  coincide  with  one  another;  this  letter 
is,  moreover,  the  initial  letter  of  the  word  Identity,  the  denomi- 
nation suitable  to  the  kind  of  relation  in  question.  It  may  also 
be  noted  that  the  three  letters  (H,  X,  I)  are  symmetrical, 
like  the  relations  they  are  intended  to  recall,  so  that  they 
are  not  liable  to  change  their  appearance  by  being  reversed. 
But  this  is  not  the  case  with  the  letter  (67)  which  on  being 
reversed  changes  into  (j);  hence  we  have  reserved  this  letter 
to  recall  a  relation  in  which  the  two  ideas  play  different  parts, 
a  relation  which  is  not  at  all  reciprocal.  This  letter  is,  moreover, 
the  initial  letter  common  to  both  of  the  words  Containing 
and  Contained,  which  well  express  the  relative  situation  of  the 
two  ideas. 

But  it  was  Charles  S.  Peirce  who  in  1870  systematically  elaborated 
the  concept  of  inclusion. 

40.13  Inclusion  in  or  being  as  small  as  is  a  transitive  relation. 
The  consequence  holds  that 

If      x-<y, 

and  y  — <  z, 

then  x  — <  z. 
(Footnote)  I  use  the  sign  — <  in  place  of  ^.  My  reasons  for 
not  liking  the  latter  sign  are  that  it  cannot  be  written  rapidly 
enough,  and  that  it  seems  to  represent  the  relation  it  expresses 
as  being  compounded  of  two  others  which  in  reality  are 
complications  of  this.  It  is  universally  admitted  that  a  higher 
conception  is  logically  more  simple  than  a  lower  one  under  it. 
Whence  it  follows  from  the  relations  of  extension  and  com- 
prehension, that  in  any  state  of  information  a  broader  concept 
is  more  simple  than  a  narrower  one  included  under  it.  Now  all 
equality  is  inclusion  in,  but  the  converse  is  not  true;  hence 



inclusion  in  is  a  wider  concept  than  equality,  and  therefore 
logically  a  simpler  one.  On  the  same  principle,  inclusion  is  also 
simpler  than  being  less  than.  The  sign  ^  seems  to  involve  a 
definition  by  enumeration;  and  such  a  definition  offends 
against  the  laws  of  definition. 

Schroder  introduces  and  explains  the  symbol  of  inclusion  from 
the  start: 

40.14  Examples  of  categorical  judgements  of  the  simplest 
kind  are  propositions  accepted  as  true  in  chemistry: 

'Gold  is  metal'  -  'Common  salt  is  sodium  chloride'.  - 

Even  to  these  we  can  very  easily  link  the  basic  contrasts 
needed  in  our  science. 

Both  statements  have  the  same  copula.  .  .  .  Yet  the  factual 
relation  between  the  subject  and  predicate  of  the  statement  is 
essentially  different  in  the  first  and  in  the  second  case,  insofar 
as  conversely  metal  is  not  always  gold,  while,  sodium  chloride  is 
also  common  salt.  This  difference  is  not  expressed  in  a  way 
apparent  to  the  eye  in  the  original  statements. 

If  it  is  now  desired  to  exhibit  the  factual  relation  between 

subject  and  predicate  by  a  relative  symbol  more  exactly  than 

those  statements  do,  a  symbol  must  be   chosen  for  the   first 

example  different  from  that  for  the  second.  One  might  write : 

gold  (^  metal  common  salt   =  sodium  chloride 

40.15  The  other  symbol  Q  can  be  read  .  .  .  'subordinated'.  It 
is  called  the  symbol  of  subordination  and  a  statement  such  as 

a  'subordination'.  The  symbol  is  shaped  similarly  to,  and  to 
some  extent  in  imitation  of,  the  'inequality  symbol'  of 
arithmetic,  viz.  the  symbol  <  for  'less  [than]'.  As  is  well 
known,  this  can  be  read  backwards  as  'greater',  >,  and  it  is 
easily  impressed  on  the  memory  together  with  its  meaning  if 
one  bears  in  mind  that  the  symbol  broadens  from  the  smaller 
to  the  larger  value,  or  points  from  the  larger  value  towards 
the  smaller.  Analogously,  our  symbol  of  subordination,  when 
read  backwards  in  the  reversed  position,  ^),  i.e.  reading  again 
from  left  to  right,  will  mean  '  super  ordinate  d\  The  original 
subordination  may  also  be  written  backwards  as  a  superordi- 
nation' : 

and  this  expression  means  just  the  same  as  the  original  one. 



40.16  The  copula  'is'  is  sometimes  used  to  express  one, 
sometimes  the  other  of  the  relations  which  we  have  shown 
by  means  of  the  symbols  (2  and  =.  For  its  exhibition  a  symbol 
composed  of  both  the  two  last,  =£,  is  chiefly  recommended,  as 
being  immediately,  and  so  to  say  of  itself,  intelligible,  and 
readily  memorizable.  In  fullest  detail,  this  symbol  is  to  be 
read  as  'subordinated  or  equaV .  .  .  . 

A  statement  of  the  form 

is  called  a  sub  sumption,  the  symbol  =£[  the  symbol  of  sub- 


The  term  of  this  whole  development  is  to  be  found  in  the  sym- 
bolism which  Giuseppe  Peano  published  in  1889.  This  comprises 
essentially  more  than  the  Boolean  calculus  and  at  the  same  time 
brings  the  latter  to  its  final  form.  Its  essentials  will  be  given  below 




We  speak  first  of  the  development  of  proposition-determining 

functors  and  other  fundamental  parts  of  proposition;! I  Logic.  This 
was  first  formulated,  in  the  modern  period  of  Logic,  by  Boole  - 
actually  as  the  second  possible  interpretation  of  his  calculus  (1847  . 
A  more  exact  exposition  appears  in  McColl  (1877).  Frege's  Begrif/s- 
schrift  (1879)  marks  a  new  beginning,  in  this  as  in  so  many  other 
regions  of  formal  logic.  In  connection  with  Frege's  doctrine  of 
implication  we  give  also  two  important  texts  from  Peirce. 

Later,  Peano  (1889)  introduced  a  symbolism  which  is  notably 
easier  to  read  than  Frege's;  Russell's  displays  only  inessential 
variations  from  it.  But  the  symbolism  which  Lukasiewicz  later 
constructed,  in  dependence  on  Frege,  is  basically  different  from 

We  read  in  the  Analysis: 

41.01   Of  the  conditional  syllogism  there  are  two,  and  only 
two  formulae. 
1st     The  constructive, 

If  A  is  B,  then  C  is  D, 

But  A  is  B,  therefore  C  is  D. 
2nd    The  destructive, 

If  A  is  B,  then  C  is  D, 

But  C  is  not  D,  therefore  A  is  not  B. 
...  If  we  examine  either  of  the  forms  of  conditional  syllog- 
ism above  given,  we  shall  see  that  the  validity  of  the  argument 
does  not  depend  upon  any  considerations  which  have  reference 
to  the  terms  A,  B,  C,  D,  considered  as  the  representatives  of 
individuals  or  of  classes.  We  may,  in  fact,  represent  the 
Propositions  A  is  B,  C  is  D,  by  the  arbitrary  symbols  X  and  Y 
respectively,  and  express  our  syllogisms  in  such  forms  as  the 
following : 

If  X  is  true,  then  Y  is  true, 

But  X  is  true,  therefore  Y  is  true. 
Thus,  what  we  have  to  consider  is  not  objects  and  classes 
of  objects,  but  the  truths  of  Propositions,  namely,  of  those 
elementary  Propositions  which  are  embodied  in  the  terms  of 
our  hypothetical  premises. 



41.02  If  we  confine  ourselves  to  the  contemplation  of  a 
given  proposition  X,  and  hold  in  abeyance  every  other  conside- 
ration, then  two  cases  only  are  conceivable,  viz.  first  that  the 
given  Proposition  is  true,  and  secondly  that  it  is  false.  As 
these  cases  together  make  up  the  Universe  of  the  Proposition, 
and  as  the  former  is  determined  by  the  elective  symbol  x, 
the  latter  is  determined  by  the  symbol  1  -  x. 

But  if  other  considerations  are  admitted,  each  of  these 
cases  will  be  resolvable  into  others,  individually  less  extensive, 
the  number  of  which  will  depend  upon  the  number  of  foreign 
considerations  admitted.  Thus  if  we  associate  the  Propositions 
X  and  y,  the  total  number  of  conceivable  cases  will  be  found 
as  exhibited  in  the  following  scheme. 

Cases  Elective  expressions 

1st     X  true,  y  true  xy 

2nd    X  true,  Y  false x  (1  -y) 

3rd     X  false,  Y  true (1  -x)  y 

4th     X  false,  Y  false (l-x){l-y). 

41.03  And  it  is  to  be  noted  that  however  few  or  many  those 
circumstances  may  be,  the  sum  of  the  elective  expressions 
representing  every  conceivable  case  will  be  unity.  Thus  let 
us  consider  the  three  Propositions.  X,  It  rains,  Y,  It  hails, 
S,  It  freezes.  The  possible  cases  are  the  following: 

Cases  Elective    expressions 

1st     It  rains,  hails,  and  freezes,  xyz 

2nd    It  rains  and  hails,  but  does  not 

freeze  xy  (1  -  z) 

3rd     It  rains  and  freezes,  but  does  not 

hail  xz  (1  -  y) 

4th     It  freezes  and  hails,  but  does  not      yz  (1  -  x) 

5th     It  rains,  but  neither  hails  nor  x  (1  -  y)  (1  -  z) 

6th     It  hails,  but  neither  rains  nor 

freezes  y  (1  -x)  (1  -  z) 

7th     It  freezes,  but  neither  hails  nor         z  (1  -  x)  (1  -  y) 

8th     It  neither  rains,  hails,  nor  freezes      (1  -  x)  (1  -  y)  (1  -  z) 

1  —  sum 



41.04  ...  To  express  that  a  given  Proposition  X  is  true. 

The  symbol  1  -  x  selects  those  cases  in  which  the  Propo- 
sition X  is  false.  But  if  the  Proposition  is  true,  there  are  no 
such  cases  in  its  hypothetical  Universe,  therefore 

1  -  x  =  0, 
or  x  =  1. 

To  express  that  a  given  Proposition  X  is  false. 

The  elective  symbol  x  selects  all  those  cases  in  which  the 
Proposition  is  true,  and  therefore  if  the  Proposition  is  false, 

x  =  6. 

These  principles  are  then  applied  just  like  those  of  the  logic  of 
classes,  to  syllogistic  practice. 

The  similarity  of  the  table  of  four  cases  in  41.02  with  Philo's 
matrix  of  truth-values  (20.07)  is  to  be  noticed.  As  has  already  been 
said,  the  Boolean  calculus  had  no  symbol  for  implication,  nor  yet 
one  for  negation;  both  are  introduced  by  means  of  more  complex 
formulae.  In  place  of  the  logical  sum,  Boole  had  the  notion  of 
exclusive  disjunction.  Hence  it  is  that  propositional  logic  is  made 
to  appear  as  a  discipline  co-ordinate  with,  if  not  subordinate  to, 
the  logic  of  classes,  by  contrast  to  the  clear  insight  possessed  by  the 
Stoics  and  Scholastics  into  its  nature  as  basic. 


Passing  over  the  development  that  occurred  between  1847  and 
1877,  mainly  due  to  Jevons  and  Peirce,  we  now  give  instead  a  text 
from  Hugh  McGoll  (1877)  in  which  propositional  logic  is  emancipated 
from  the  calculus  of  classes,  and  endowed  with  all  the  symbols  just 
mentioned.  In  a  way  this  text  marks  the  highest  level  of  mathe- 
matical logic  before  Frege. 

41.05  Definition  1.  -  Let  any  symbols,  say  A,  B,  C,  etc., 
denote  statements  [or  propositions]  registered  for  con- 
venience of  reference  in  a  table.  Then  the  equation  A  =  1 
asserts  that  the  statement  A  is  true;  the  equation  A  =  0 
asserts  that  the  statement  A  is  false ;  and  the  equation  A  =  B 
asserts  that  A  and  B  are  equivalent  statements. 

41.06  Definition  2.  -  The  symbol  A  x  B  x  C  or  ABC 
denotes  a  compound  statement,  of  which  the  statements 
A,  B,  C  may  be  called  the  factors.  The  equation  ABC  =  1 
asserts  that  all  the  three  statements  are  true;  the  equation 
ABC  =  0  asserts  that  all  the  three  statements  are  not  true, 
i.e.  that  at  least  one  of  the  three  is  false.  Similarly  a  com- 
pound statement  of  any  number  of  factors  may  be  defined. 



41.07  Definition  3.  -  The  symbol  A  +  B  +  C  denotes  an 
indeterminate  statement,  of  which  the  statements  A,  B,  C 
may  be  called  the  terms.  The  equation  A  +  B  +  C  =  0  asserts 
that  all  the  three  statements  are  false ;  the  equation  A  +  B  +  C 
=  1  asserts  that  all  the  three  statements  are  not  false,  i.e., 
that  at  least  one  of  the  three  is  true.  Similarly  an  indeterminate 
statement  of  any  number  of  terms  may  be  defined. 

41.08  Definition  4.  -  The  symbol  A'  is  the  denial  of  the 
statement  A.  The  two  statements  A  and  A'  are  so  related 
that  they  satisfy  the  two  equations  A  +  A'  =  1  and  A  A '  =  0; 
that  is  to  say,  one  of  the  two  statements  (either  A  or  A')  must 
be  true  and  the  other  false.  The  same  symbol  {i.e.  a  dash) 
will  convert  any  complex  statement  into  its  denial.  For 
example,  (AB)'  is  the  denial  of  the  compound  statement 
AB.  .  .  . 

41.09  Definition  5.  -  When  only  one  of  the  terms  of  an 
indeterminate  statement  A  +  B  +  C  +  .  .  .  can  be  true,  or 
when  no  two  terms  can  be  true  at  the  same  time,  the  terms 
are  said  to  be  mutually  inconsistent  or  mutually  exclusive. 

41.10  Definition  12.  -  The  symbol  A: B  [which  may  be  call- 
ed an  implication]  asserts  that  the  statement  A  implies  B;  or 
that  whenever  A  is  true  B  is  also  true. 

Note.  -  It  is  evident  that  the  implication  A:B  and  the 
equation  A  =  AB  are  equivalent  statements. 

1.  Content  and  judgment 

A  new  period  of  propositional  logic  begins  with  Gottlob  Frege. 
His  first  work,  the  Begriffsschrift  of  1879,  already  contains  in  brief 
an  unusually  clear  and  thorough  presentation  of  a  long  series  of 
intuitions  unknown  to  his  immediate  predecessors,  while  those 
already  familiar  are  better  formulated.  To  start  with  we  choose  a 
text  relevant  rather  to  semantics  than  logic,  in  which  this  great 
thinker  introduces  of  his  propositional  logic  with  the  'judgment- 
stroke'  : 

41.11  A  judgment  is  always  to  be  expressed  by  means  of 
the  sign 

This  stands  to  the  left  of  the  sign  or  complex  of  signs  in  which 
the  content  of  the  judgment  is  given.  If  we  omit  the  little 
vertical  stroke  at  the  left  end  of  the  horizontal  stroke,  then 



the  judgment  is  to  be  transformed  into  a  mere  complex  of 
ideas;  the  author  is  not  expressing  his  recognition  or  non- 
recognition  of  the  truth  of  this.  Thus,  let 

I— A 
mean    the    judgment:    'unlike    magnetic    poles    attract    one 
another'.  In  that  case 


will  not  express  this  judgment;  it  will  be  intended  just  to 
produce  in  the  reader  the  idea  of  the  mutual  attraction  of 
unlike  magnetic  poles  -  so  that,  e.g.,  he  may  make  inferences 
from  this  thought  and  test  its  correctness  on  the  basis  of 
these.  In  this  case  we  qualify  the  expression  with  the  words 
'the  circumstance  thaf  or  'the  proposition  lhal\ 

Not  every  content  can  be  turned  into  a  judgment  by 
prefixing  | —  to  a  symbol  for  the  content;  e.g.,  the  idea  'house' 
cannot.  Hence  we  distinguish  contents  that  are,  and  contents 
that  are  not,  possible  contents  of  judgment. 

As  a  constituent  of  the  sign  | —  the  horizontal  stroke  combines 
the  symbols  following  it  into  a  whole;  assertion,  which  is  expressed 
by  the  vertical  stroke  at  the  left  end  of  the  horizontal  one,  relates 
to  the  whole  thus  formed.  The  horizontal  stroke  I  wish  to  call 
the  content-stroke,  and  the  vertical  the  judgment-stroke.  The 
content-stroke  is  also  to  serve  the  purpose  of  relating  any  sign 
whatsoever  to  the  whole  formed  by  the  symbols  following 
the  stroke.  The  content  of  what  follows  the  content-stroke  must 
always  be  a  possible  content  of  judgment. 

2.  Implication 

Frege  then  introduces  the  Philonian  concept  of  implication, 
though,  unlike  Peirce  (41.14)  he  knows  nothing  in  this  connection 
of  Philo  or  the  Scholastics.  It  is  remarkable  that  he  proceeds  almost 
exactly  like  Philo. 

41.12  If  A  and  B  stand  for  possible  contents  of  judgment, 
we  have  the  four  following  possibilities : 
(i)  A  affirmed,  B  affirmed; 
(ii)  A  affirmed,  B  denied; 
(iii)  A  denied,  B  affirmed; 
(iv)  A  denied,  B  denied. 

l—  B 

stands    for    the    judgment   that    the    third   possibility    is    not 



realized,  but  one  of  the  other  three  is.  Accordingly,  the  denial  of 

l—  B 
is  an  assertion  that  the  third  possibility  is  realized,  i.e.  that 
A  is  denied  and  B  affirmed. 
From  among  the  cases  where 

•—  B 

is  affirmed,  the  following  may  be  specially  emphasized: 

(1)  A  is  to  be  affirmed.  -  In  this  case  the  content  of  B  is 
quite  indifferent.  Thus,  let  i —  A  mean:  3  x  7  =  31 ;  let  £ 
stand  for  the  circumstance  of  the  sun's  shining.  Here  only 
the  first  two  cases  out  of  the  four  mentioned  above  are 
possible.  A  causal  connection  need  not  exist  between  the  two 

(2)  B  is  to  be  denied.  -  In  this  case  the  content  of  A  is 
indifferent.  E.g.,  let  B  stand  for  the  circumstance  of  perpetual 
motion's  being  possible,  and  A  for  the  circumstance  of  the 
world's  being  infinite.  Here  only  the  second  and  fourth  of  the 
four  cases  are  possible.  A  causal  connection  between  A  and  B 
need  not  exist. 

(3)  One  may  form  the  judgment 

l—  B 
without  knowing  whether  A  and  B  are  to  be  affirmed  or  denied. 
E.g.,  let  B  stand  for  the  circumstance  of  the  Moon's  being  in 
quadrature  with  the   Sun,   and  A   the   circumstance  of  her 
appearing  semi-circular.  In  this  case  we  may  render 

l—  B 

by  means  of  the  conjunction  'if;  'if  the  Moon  is  in  quadrature 
with  the  Sun,  then  she  appears  semi-circular'.  The  causal 
connection  implicit  in  the  word  'if  is,  however,  not  expressed 
by  our  symbolism;  although  a  judgment  of  this  sort  can  be 
made  only  on  the  ground  of  such  a  connection.  For  this  con- 
nection is  something  general,  and  as  yet  we  have  no  expression 
for  generality. 

The  text  needs  some  explanations.  First,  Frege  uses  A1  for  the 
consequent  and  '  B'  for  the  antecedent  -  contrary  to  ordinary  usage, 
but  like  Aristotle;  so  the  antecedent  stands  in  the  lower  place.  So 



the  schema  excludes  only  the  case  where  the  antecedent  (B)  is  true 
and  the  consequent  (A)  is  false;  in  all  the  other  three  cases  the 
proposition  is  true.  Thus  we  have  just  the  same  state  of  affairs  as  in 
Philo  (20.07):  the  schema  is  a  symbol  of  Philonian  implication.  It 
signifies  'if  B,  then  A1  in  the  Philonian  sense  of  'if. 

Important  is  the  stress  laid  on  the  fact  that  implication  has 
nothing  to  do  with  the  causal  connection  between  the  facts  signified 
by  the  antecedent  and  consequent. . 


Philonian  implication  alone  continued  to  be  used  in  mathematical 
logic  up  to  1918  -  unlike  usage  in  the  Stoic  and  Scholastic  periods. 
One  of  the  best  justifications  of  this  concept  which  seems  so  odd  to 
the  man  in  the  street,  is  to  be  found  in  a  fairly  late  text  of  Peirce's, 
dated  1902. 

41.13  To  make  the  matter  clear,  it  will  be  well  to  begin  by 
defining  the  meaning  of  a  hypothetical  proposition,  in  general. 
What  the  usages  of  language  may  be  does  not  concern  us; 
language  has  its  meaning  modified  in  technical  logical  for- 
mulae as  in  other  special  kinds  of  discourse.  The  question 
is  what  is  the  sense  which  is  most  usefully  attached  to  the 
hypothetical  proposition  in  logic  ?  Now,  the  peculiarity  of  the 
hypothetical  proposition  is  that  it  goes  out  beyond  the  actual 
state  of  things  and  declares  what  would  happen  were  things 
other  than  they  are  or  may  be.  The  utility  of  this  is  that 
it  puts  us  in  possession  of  a  rule,  say  that  'if  A  is  true,  B  is 
true',  such  that  should  we  hereafter  learn  something  of  which 
we  are  now  ignorant,  namely  that  A  is  true,  then  by  virtue 
of  this  rule,  we  shall  find  that  we  know  something  else,  namely, 
that  B  is  true.  There  can  be  no  doubt  that  the  Possible,  in  its 
primary  meaning,  is  that  which  may  be  true  for  aught  we 
know,  that  whose  falsity  we  do  not  know.  The  purpose  is 
subserved,  then,  if  throughout  the  wrhole  range  of  possibility, 
in  every  state  of  things  in  which  A  is  true,  B  is  true  too. 
The  hypothetical  proposition  may  therefore  be  falsified  by  a 
single  state  of  things,  but  only  by  one  in  which  A  is  true 
while  B  is  false.  States  of  things  in  which  A  is  false,  as  well 
as  those  in  which  B  is  true,  cannot  falsify  it.  If,  then,  B  is  a 
proposition  true  in  every  case  throughout  the  whole  range  of 
possibility,  the  hypothetical  proposition,  taken  in  its  logical 
sense,  ought  to  be  regarded  as  true,  whatever  may  be  the 



usage  of  ordinary  speech.  If,  on  the  other  hand,  A  is  in  no 
case  true,  throughout  the  range  of  possibility,  it  is  a  matter 
of  indifference  whether  the  hypothetical  be  understood  to 
be  true  or  not,  since  it  is  useless.  But  it  will  be  more  simple 
to  class  it  among  true  propositions,  because  the  cases  in  which 
the  antecedent  is  false  do  not,  in  any  other  case,  falsify  a 
hypothetical.  This,  at  any  rate,  is  the  meaning  which  I  shall 
attach  to  the  hypothetical  proposition  in  general,  in  this 

Also  of  interest  is  the   following  remark   of  the  same  logician 


41.14  Although  the  Philonian  views  lead  to  such  incon- 
veniences as  that  it  is  true,  as  a  consequence  de  inesse,  that  if 
the  Devil  were  elected  president  of  the  United  States,  it 
would  prove  highly  conducive  to  the  spiritual  welfare  of  the 
people  (because  he  will  not  be  elected),  yet  both  Professor 
Schroder  and  I  prefer  to  build  the  algebra  of  relatives  upon 
this  conception  of  the  conditional  proposition.  The  incon- 
venience, after  all,  ceases  to  seem  important,  when  we  reflect 
that,  no  matter  what  the  conditional  proposition  be  under- 
stood to  mean,  it  can  always  be  expressed  by  a  complexus  of 
Philonian  conditionals  and  denials  of  conditionals. 


Some  examples  of  the  applications  of  Frege's  implication-schema 
will  make  his  main  ideas  clearer. 

41.15  The  vertical  stroke  joining  the  two  horizontal  ones 
is  to  be  called  the  conditional-stroke.  .  .  .  Hence  it  is  easy  to 
see  that 

I rA 

]—  B 

—  r 

denies  the  case  in  which  A  is  denied,  B  and  V  are  affirmed. 
This  must  be  thought  of  as  compounded  of 

,4  and/1 

just  as 



is  from  A  and  B.  Thus  we  first  have  the  denial  of  the  case 
in  which 

is  denied,  Y  is  affirmed.  But  the  denial  of 

signifies  that  A  is  denied  and  B  is  affirmed.  Thus  we  obtain 
what  is  given  above. 

41.16  From  the  explanation  given  in  §  5  (41.12)  it  is  obvious 
that  from  the  two  judgments 

| — , —  A  and  I —  B 

there  follows   the  new  judgment   i —   A.   Of  the  four  cases 
enumerated  above,  the  third  is  excluded  by 


and  the  second  and  fourth  by : 

\—  B, 
so  that  only  the  first  remains. 

41.17  Let  now  X  for  example  signify  the  judgment 

I—  B 

—  or  one  which  j — , —  A  contains  as  a  particular  case.  Then 
I—  B 

I  write  the  inference  thus: 

\—  B 


I—  A. 
Here  it  is  left  to  the  reader  to  put  together  the  judgment 

]—  B 

from  | —  B  and  | —  A,  and  see  that  it  tallies  with  the  cited 
judgment  X. 



Frege  uses  the  same  schemata,  together  with  the  'negation-stroke', 
to  express  the  logical  sum. 

41.18  If  a  small  vertical  stroke  is  attached  to  the  lower 
side  of  the  content-stroke,  this  shall  express  the  circumstance 
of  the  content's  not  being  the  case.  Thus,  e.g.,  the  meaning 

h-r-  A: 
is:  'A  is  not  the  case'.  I  call  this  small  vertical  stroke  the 

41.19  We  now  deal  with  some  cases  where  the  symbols 
of  conditionality  and  negation  are  combined. 


means :  'the  case  in  which  B  is  to  be  affirmed  and  the  negation 
of  A  is  to  be  denied  does  not  occur';  in  other  words,  'the 
possibility  of  affirming  both  A  and  B  does  not  exist',  or  A  and 
B  are  mutually  exclusive'.  Thus  only  the  three  following 
cases  remain: 

A  affirmed,  B  denied; 

A  denied,  B  affirmed; 

A  denied,  B  denied. 
From  what  has  already  been  said,  it  is  easy  to  determine 
the  meaning  possessed   by  each  of  the  three   parts  of  the 
horizontal  stroke  preceding  A. 

means:  'the  case  in  which  A  is  denied  and  negation  of  B  is 
affirmed  does  not  exist';  or,  lA  and  B  cannot  both  be  denied'. 
There  remain  only  the  following  possibilities : 

A  affirmed,  B  affirmed; 

A  affirmed,  B  denied; 

A  denied,  B  affirmed. 
A  and   B  between  them  exhaust  all  possibilities.  Now  the 
words  'or',  'either  -  or',  are  used  in  two  ways.   In  its  first 

means  just  the  same  as 

A  or  B' 



i.  e.  that  nothing  besides  A  and  B  is  thinkable.  E.g.,  if  a 
gaseous  mass  is  heated,  then  either  its  volume  or  its  pressure 
increases.  Secondly,  the  expression 

lA  or  B' 
may  combine  the  meaning  of 

1     F"  A  and  that  of  [~T~  A 
—  B  4"  B 

so  that  (i)  there  is  no  third  possibility  besides  A  and  B,  (ii)  A 

and  B  are  mutually  exclusive.  In  that  case  only  the  following 

two  possibilities  remain  out  of  the  four: 

A  affirmed,  B  denied; 

A  denied,  B  affirmed. 
Of  these  two  uses  of  the  expression  'A  or  J3'  the  more  impor- 
tant is  the  first,  which  does  not  exclude  the  coexistence  of  A 
and  B;  and  we  shall  use  the  word  lor'  with  this  meaning.  Perhaps 
it  is  suitable  to  distinguish  between  'or'  and  'either  -  or', 
regarding  only  the  latter  as  having  the  subsidiary  meaning 
of  mutual  exclusion. 


Frege's  symbolism  has  the  unusual  feature  of  being  two-dimen- 
sional. In  that  it  diverges  from  the  historical  practice  of  mankind 
which  has  almost  always  expressed  its  thoughts  in  one-dimensional 
writing.  It  must  be  admitted  that  this  revolutionary  novelty  has 
much  to  be  said  for  it  -  it  notably  widens  the  expressive  possibilities 
of  writing.  But  this  was  too  revolutionary;  Frege's  symbolism  did 
not  prove  generally  intelligible,  and  the  subsequent  development  took 
place  in  another  direction.  Schroder  made  no  reference  to  it  in  1892, 
Russell  admitted  in  1903  that  he  had  learned  much  from  Frege  when 
he  had  met  his  system,  but  not  having  known  it  he  followed  Peano. 
Modern  mathematical  logic,  though  its  authors  have  less  depth  of 
thought  than  Frege,  has  adopted  Peano's  symbolism.  For  this 
reason  we  quote  a  text  from  Peano's  Arithmetices  Principia  (1889)  in 
which  he  lays  down  this  intuitively  clear  and  meaningful  symbolism 
for  propositional  logic. 

41.20   I.  Concerning  punctuation 

By  the  letters  a,  b,  ...  x,  y,  ...  x',  y'  ...  we  indicate  any 
undetermined  beings.  Determined  beings  we  indicate  by  the 
signs  or  letters  P,  K,  N.  .  .  . 

For  the  most  part  we  shall  write  signs  on  one  and  the  same 
line.  To  make  clear  the  order  in  which  they  are  to  be  con- 
joined we  use  parentheses  as  in   algebra,  or  dots  .:/.::  etc. 



That  a  formula  divided  by  dots  may  be  understood,  first 
the  signs  which  are  separated  by  no  dots  are  to  be  collected, 
afterwards  those  separated  by  one  dot,  then  those  by  two 
dots,  etc. 

E.g.  let  a,  6,  c, . . .  be  any  signs.  Then  ab  •  cd  signifies  (ab)(cd)  ; 
and  ab  .  cd  :  ef  .  gh  .-.  k  signifies  (((ab)  (cd))  ((ef)  (gh)))  k. 

Signs  of  punctuation  may  be  omitted  if  there  are  formulae 
with  different  punctuation  but  the  same  sense;  or  if  only 
one  formula,  and  that  the  one  we  wish  to  write,  has  the 

To  avoid  danger  of  ambiguity  we  never  make  use  of  .  :  as 
signs  of  arithmetical  operations. 

The  only  form  of  parentheses  is  ().  If  dots  and  paren- 
theses occur  in  the  same  formula,  signs  contained  in  paren- 
theses are  to  be  collected  first. 

//.  Concerning  propositions 

By  the  sign  P  is  signified  a  proposition. 

The  sign  r»  is  read  and.  If  a,  b  are  propositions;  then  a  r»  b 
is  the  simultaneous  affirmation  of  the  propositions  a,  b.  For 
the  sake  of  brevity  we  shall  commonly  write  ab  in  place 
of  a  n  b. 

The  sign  -  is  read  not.  Let  abeaP;  then  -a  is  the  negation 
of  the  proposition  a. 

The  sign  w  is  read  or  (vet).  Let  a,  b  be  propositions;  then 
a  v  b  is  the  same  as  -  \-a.-b. 

[By  the  sign  V  is  signified  verum  or  identity;  but  we  never 
use  this  sign.] 

The  sign  A  signifies  falsum  or  absurdum. 

[The  sign  C  signifies  is  a  consequence;  thus  bCa  is  read  b 
is  a  consequence  of  the  proposition  a.  But  we  never  use  this 

The  sign  j  signifies  is  deduced  (deducitur) :  thus  a  j  b  signi- 
fies the  same  as  bCa. 


Peano's  successors  introduced  only  minor  alterations  to  his 
symbolism.  First  Russell  (1903),  who  writes  'v'  instead  of  'w',  and 
l~  instead  of  '-',  then  Hilbert  and  Ackermann  (1928)  who  write 
a  stroke  over  a  letter  for  negation,  and   'oo'   for  the  equivalence- 



sign  '  =  '  of  Frege  and  Russell.  M.  H.  ShefTer  (1928)  introduced  '|'  as  a 
sign  for  'not  both'. 

The  Polish  school,  on  the  other  hand,  developed  two  symbolic 
languages  essentially  different  from  Peano's;  those  of  St.  Lesniewski 
and  J.  Lukasiewicz.  We  shall  not  go  into  the  first,  which  is  peculiar 
and  little  used,  but  the  symbolism  of  Lukasiewicz  deserves  brief 
exposition,  both  for  its  originality  and  its  exactness.  The  essential 
feature  is  that  all  predicates  (called  by  Lukasiewicz  'functors'; 
stand  in  front  of  their  arguments;  thus  all  brackets  and  dots  are 
dispensed  with,  without  any  ambiguity  arising. 

The  various  sets  of  symbols  may  be  compared  thus: 
McColl  A'       +         x         :      '    = 
























D    [i.e.NK) 

Thus  Lukasiewicz  writes  'Cpq'  for  'p  d  q\  and  lApq'  for  'p  v  q' . 
An  example  of  a  more  complex  formula  is  'CCpqCNqNp'  instead 
of  'p  D  q  .  D  .  ^  q  D  r***  p\ 


While  nearly  everything  mentioned  so  far  is  within  the  scope  of 
Stoic  and  Scholastic  logic  -  though  the  new  logicians  knew  hardly 
anything  of  the  achievements  of  their  predecessors,  -  the  concepts  of 
function,  variable,  and  truth-value,  without  effecting  anything  radi- 
cally new,  yet  produce  so  marked  a  development  of  the  old  concept 
of  logical  form  as  to  deserve  distinct  and  thorough  treatment. 

After  an  introductory  quotation  from  De  Morgan  (1858)  con- 
cerning logical  form  in  general,  we  give  Frege's  fundamental  text  on 
the  concept  of  function  (1893),  the  explanation  and  development  of 
Frege's  thought  from  Russell  and  the  Principia  (1903  and  1910), 
and  finally  the  extension  of  the  concept  of  function  to  many-place 
functions  by  Peirce  (1892)  and  Frege  (1893).  For  the  doctrine  of 
the  variable  in  mathematical  logic  we  have  a  quotation  from  Frege's 
Begriffsschrift  (1879)  and  Russell's  elaboration  of  the  ideas  therein 
(1903  and  1910).  As  to  truth-values,  two  texts  from  Frege  (1893) 
and  one  from  Peirce  (1885)  are  to  hand. 

In  conclusion  we  exhibit  some  examples  of  modern  truth-matrices 
(truth-tables,  tables  of  truth-values),  by  which  propositional  func- 
tors are  defined,  taking  these  from  Peirce  (1902)  and  Wittgenstein 
(1921);  the  decision  procedure  based  on  them  is  illustrated  from 
Kotarbinski  (1929). 




An  important  text  of  De  Morgan's  make  a  fitting  start.  It  dates 
from  1858,  and  shows  a  very  clear  idea  of  logical  form.  It  may  be 
compared  with  Buridan's  definition  of  logical  form  (26.12):  the 
thought  is  the  same,  but  more  developed,  in  that  abstraction  is  made 
from  the  sense  of  the  logical  constants. 

42.01  In  the  following  chain  of  propositions,  there  is 
exclusion  of  matter,  form  being  preserved,  at  every  step:  - 

(Positively  true)  Every  man  is  animal 

Every  man  is  Y  Y  has  existence 

Every  X  is  Y  X  has  existence 

Every  X Y         is  a  transitive 


a  of  X Y  a  a  fraction  <  or  =  1 

(Probability  p)     (3  of  X Y  pa  fraction  <  or  =  1 

The  last  is  nearly  the  purely  formal  judgment,  with  not  a 
single  material  point  about  it,  except  the  transitiveness  of  the 

copula.  But  'is'  is  more  intense  than  the  symbol ,  which 

means  only  transitive  copula:  for  'is'  has  transitiveness,  and 
more.  Strike  out  the  word  transitive,  and  the  last  line  shews  the 
pure  form  of  the  judgment. 

The  foregoing  table  is  to  be  understood  in  the  sense  that  the 
conditions  formulated  in  one  row,  hold  for  all  subsequent  rows;  thus, 

e.g.,  the  relation  shown  by  the  stroke  (' ')  in  the  two  last  rows 

must  be  transitive,  since  this  is  laid  down  in  the  preceding  row. 

Neither  De  Morgan  nor  any  other  logician  can  remain  at  so  high 
a  level  of  abstraction  as  is  here  achieved.  Basically,  this  is  a  re- 
discovery of  the  scholastic  concept  of  form,  made  through  a  broaden- 
ing of  the  mathematical  concept  of  function,  for  which  we  refer  to 
Peirce  (42.02)  and  Frege. 


We  now  give  Frege's  fundamental  text  (1893): 

42.03  If  we  are  asked  to  state  what  the  word  'function'  as 
used  in  mathematics  originally  stood  for,  we  easily  fall  into 
saying  that  a  function  of  x  is  an  expression  formed,  by  means 
of  the  notations  for  sum,  product,  power,  difference,  and  so 
on,  out  of  V  and  definite  numbers.  This  attempt  at  a 
definition  is  not  successful  because  a  function  is  here  said  to 



be  an  expression,  a  combination  of  signs,  and  not  what  the 
combination  designates.  Accordingly  another  attempt  would 
be  made:  we  could  try  'reference  of  an  expression'  instead  of 
'expression'.  There  now  appears  the  letter  V  which  indicates 
a  number,  not  as  the  sign  '2'  does,  but  indefinitely.  For 
different  numerals  which  we  put  in  the  place  of  V,  we  get,  in 
general,  a  different  reference.  Suppose,  e.g.,  that  in  the 
expression  '(2  +  3  •  x2)  x\  instead  of  '#'  we  put  the  number- 
signs  '0',  '1',  '2',  '3',  one  after  the  other;  we  then  get  corres- 
pondingly as  the  reference  of  the  expression  the  numbers 
0,  5,  28,  87.  Not  one  of  the  numbers  so  referred  to  can  claim 
to  be  our  function.  The  essence  of  the  function  comes  out 
rather  in  the  correspondence  established  between  the  numbers 
whose  signs  we  put  for  V  and  the  numbers  which  then  appear 
as  the  reference  of  our  expression  -  a  correspondence  which  is 
represented  intuitively  by  the  course  of  the  curve  whose 
equation  is,  in  rectangular  co-ordinates,  'y  =  (2  +  3.  x2)  x\  In 
general,  then,  the  essence  of  the  function  lies  in  the  part  of 
the  expression  which  is  there  over  and  above  the  lx\  The 
expression  of  a  function  needs  completion,  is  'unsaturated' .  The 
letter  lx'  only  serves  to  keep  places  open  for  a  numerical  sign 
to  be  put  in  and  complete  the  expression;  and  thus  it  enables 
us  to  recognize  the  special  kind  of  need  for  a  completion  that 
constitutes  the  peculiar  nature  of  the  function  symbolized 
above.  In  what  follows,  the  Greek  letter  %'  will  be  used 
instead  of  the  letter  V.  This  'keeping  open'  is  to  be  understood 
in  this  way:  All  places  in  which  '£'  stands  must  always  be 
filled  by  the  same  sign  and  never  by  different  ones.  I  call  these 
places  argument-places,  and  that  whose  sign  or  name  takes 
these  places  in  a  given  case  I  call  the  argument  of  the  function 
for  this  case.  The  function  is  completed  by  the  argument: 
I  call  what  it  becomes  on  completion  the  value  of  the  function 
for  the  argument.  We  thus  get  a  name  of  the  value  of  a  func- 
tion for  an  argument  when  we  fill  the  argument-places  in  the 
name  of  the  function  with  the  name  of  the  argument.  Thus, 
e.g.,  '(2  +  3  .  I2)  1'  is  name  of  the  number  5,  composed  of  the 
function-name  '(2  +  3  .  £2)  £'  and  '1'.  The  argument  is  not  to 
be  reckoned  in  with  the  function,  but  serves  to  complete  the 
function,  which  is  'unsaturated'  by  itself.  When  in  the  sequel 
an  expression  like  'the  function  O  (£)'  is  used,  it  is  always  to  be 
observed  that  the  only  service  rendered  by  '£'  in  the  symbol 
for  the  function  is  that  it  makes  the  argument-places  recogniz- 



able;   it   does   not  imply  that  the   essence   of  the  function 
becomes  changed  when  any  other  sign  is  substituted  for  '£'. 

The  following  remarks  will  assist  understanding  of  this  pioneer 
passage.  In  mathematical  usage  the  word  'function'  has  two  refer- 
ences, usually  not  very  clearly  distinguished.  On  the  one  hand  it 
stands  for  an  expression  (formula)  in  which  a  variable  occurs,  on  the 
other  for  the  'correspondence  between  numbers'  for  which  such  an 
expression  stands,  and  so  for  some  kind  of  lecton  or  in  general,  for 
that  for  which  the  expression  stands,  (which  in  any  case  is  not  a 
written  symbol).  Frege  makes  a  sharp  distinction  between  these  two 
references,  and  allows  only  the  second  to  the  word  'function'  - 
conformably  with  his  general  position  that  logic  (and  mathematics) 
has  as  its  object  not  symbols  but  what  they  stand  for.  It  is  important 
to  understand  this,  because  Russell  and  nearly  all  logicians  after  him 
will  speak  of  expressions  and  formulae  as  'functions',  unlike  Frege. 

However,  this  opposition  is  irrelevant  to  the  basic  logical  problems 
considered  here.  Frege,  too,  makes  use  of  analysis  of  expressions  to 
convey  his  thought,  and  what  he  states  in  the  text  just  quoted, 
holds  good  for  every  interpretation  of  the  word  'function'.  He  intro- 
duces, namely,  three  fundamental  concepts:  1.  of  the  argument  and 
argument-place,  2.  of  a  value,  3.  of  an  'unsaturated'  function,  i.e. 
one  containing  a  variable. 


Russell  who  knew  the  work  of  Frege  well,  followed  his  ideas  but 
with  some  divergences.  He  seems  to  start  from  the  Aristotelian 
concept  of  proposition  rather  than  from  the  mathematical  concept 
of  function,  and  as  already  said,  apparently  interprets  the  word 
'function'  as  the  name  of  an  expression  or  written  formula.  In  the 
Principles  (1903)  he  writes: 

42.04  It  has  always  been  customary  to  divide  propositions 
into  subject  and  predicate;  but  this  division  has  the  defect 
of  omitting  the  verb.  It  is  true  that  a  graceful  concession  is 
sometimes  made  by  loose  talk  about  the  copula,  but  the  verb 
deserves  far  more  respect  than  is  thus  paid  to  it.  We  may  say, 
broadly,  that  every  proposition  may  be  divided,  some  in 
only  one  way,  some  in  several  ways,  into  a  term  (the  subject) 
and  something  which  is  said  about  the  subject,  which  some- 
thing I  shall  call  the  assertion.  Thus  'Socrates  is  a  man'  may 
be  divided  into  Socrates  and  is  a  man.  The  verb,  which  is  the 
distinguishing  mark  of  propositions,  remains  with  the  asser- 
tion; but  the  assertion  itself,  being  robbed  of  its  subject,  is 
neither  true  nor  false.  .  .  . 



If  this  text  is  compared  with  12.01  and  similar  passages  in  Aris- 
totle, it  can  be  seen  that  Russell  here  opts  for  the  original  Aristotelian 

analysis  of  propositions  against  that  of  the  later  'classical'  logic. 
In  this  connection  he  seems  to  have  been  the  first  to  formulate 
expressly  the  idea  that  when  the  subject  is  replaced  by  a  variable,  the 
resulting  formula  -  the  propositional  function  is  no  longer  a 
proposition.  The  same  problem  is  still  more  explicitly  treated  in  the 

42.05  By  a  'propositional  function'  we  mean  something 
which  contains  a  variable  x,  and  expresses  a  proposition  as 
soon  as  a  value  is  assigned  to  x.  That  is  to  say,  it  differs  from 
a  proposition  solely  by  the  fact  that  it  is  ambiguous:  it 
contains  a  variable  of  which  the  value  is  unassigned.  .  .  . 

42.06  The  question  as  to  the  nature  of  a  function  is  by  no 
means  an  easy  one.  It  would  seem,  however,  that  the  essential 
characteristic  of  a  function  is  ambiguity.  Take,  for  example, 
the  law  of  identity  in  the  form  lA  is  A\  which  is  the  form  in 
which  it  is  usually  enunciated.  It  is  plain  that,  regarded 
psychologically,  we  have  here  a  single  judgment.  But  what 
are  we  to  say  of  the  object  of  judgment?  We  are  not  judging 
that  Socrates  is  Socrates,  nor  that  Plato  is  Plato,  nor  any 
other  of  the  definite  judgments  that  are  instances  of  the  law 
of  identity.  Yet  each  of  these  judgments  is,  in  a  sense,  within 
the  scope  of  our  judgment.  We  are  in  fact  judging  an  ambi- 
guous instance  of  the  propositional  function  'A  is  A'.  We  appear 
to  have  a  single  thought  which  does  not  have  a  definite 
object,  but  has  as  its  object  an  undetermined  one  of  the  values 
of  the  function  'A  is  A'.  It  is  this  kind  of  ambiguity  that 
constitutes  the  essence  of  a  function.  When  we  speak  of  lyx', 
where  x  is  not  specified,  we  mean  one  value  of  the  function, 
but  not  a  definite  one.  We  may  express  this  by  saying  that  'yx' 
ambiguously  denotes  cpa,  96,  cpc,  etc.,  where  cpa,  96,  9c,  etc. 
are  the  various  values  of  lyx'. 


Perhaps  even  more  important  than  the  broadening  of  the  concept 
of  function  to  include  non-mathematical  domains,  is  the  extension  to 
many-place  functions  achieved  by  Frege  and  Peirce.  The  resulting 
extension  of  the  Aristotelian  subject-predicate  schema  is  something 
quite  new  in  formal  logic.  Our  first  text  is  Peirce's  (1892) : 

42.07  If  upon  a  diagram  we  mark  two  or  more  points  to 
be  identified  at  some  future  time  with  objects  in  nature,  so  as 



to  give  the  diagram  at  that  future  time  its  meaning;  or  if  in 
any  written  statement  we  put  dashes  in  place  of  two  or  more 
demonstratives  or  pro-demonstratives,  the  professedly  incom- 
plete representation  resulting  may  be  termed  a  relative  rhema. 
It  differs  from  a  relative  term  only  in  retaining  the  'copula',  or 
signal  of  assertion.  If  only  one  demonstrative  or  pro-demon- 
strative  is   erased,   the   result  is   a   non-relative   rhema.   For 

example,  ' buys from for  the  price  of ',  is  a 

relative  rhema;  it  differs  in  a  merely  secondary  way  from 

' is  bought  by from for ' , 

from  ' sells  ---  to for ', 

and  from  ' is  paid  by to for '. 

On  the  other  hand,  ' is  mortal'  is  a  non-relative  rhema. 

42.08  A  rhema  is  somewhat  closely  analogous  to  a  chemical 
atom  or  radicle  with  unsaturated  bonds.  A  non-relative 
rhema  is  like  a  univalent  radicle;  it  has  but  one  unsaturated 
bond.  A  relative  rhema  is  like  a  multivalent  radicle.  The 
blanks  of  a  rhema  can  only  be  filled  by  terms,  or,  what  is  the 
same  thing,  by  'something  which'  (or  the  like)  followed  by  a 
rhema;  or,  two  can  be  filled  together  by  means  of  'itself  or 
the  like.  So,  in  chemistry,  unsaturated  bonds  can  only  be 
saturated  by  joining  two  of  them,  which  will  usually,  though 
not  necessarily,  belong  to  different  radicles.  If  two  univalent 
radicles  are  united,  the  result  is  a  saturated  compound.  So, 
two  non-relative  rhemas  being  joined  give  a  complete  propo- 
sition. Thus,  to  join  ' is  mortal'  and 'is  a  man',  we 

have  'A  is  mortal  and  AT  is  a  man',  or  some  man  is  mortal.  So 
likewise,  a  saturated  compound  may  result  from  joining  two 
bonds  of  a  bivalent  radicle;  and,  in  the  same  way,  the  two 
blanks  of  a  dual  rhema  may  be  joined  to  make  a  complete 

proposition.  Thus,  ' loves ',  'A  loves  A',  or  something 

loves  itself. 

Frege,  a  year  later,  writes  in  the  same  sense: 

42.09  So  far  we  have  only  spoken  of  functions  of  one 
argument;  but  we  can  easily  make  the  transition  to  func- 
tions with  two  arguments.  These  need  a  double  completion  in 
that  after  a  completion  by  one  argument  has  been  effected,  a 
function  with  one  argument  is  obtained.  Only  after  another 
completion  do  we  reach  an  object,  which  is  then  called  the 
value  of  the  function  for  the  two  arguments.  Just  as  we  made 



use  of  the  letter  '£'  for  functions  with  one  argument,  so  we  now 
use  the  letters  '£'  and  %'  to  express  the  twofold  unsaturatedness 
of  functions  with  two  arguments,  as  in  '(<;  +  £)2  +  £\  In 
substituting,  e.g.,  '1'  for  '£'  we  saturate  the  function  to  the 
extent  that  in  (§  +  l)2  +  1  we  are  left  with  a  function  with 
only  one  argument.  This  way  of  using  the  letters  '£'  and  '£' 
must  always  be  kept  in  view  when  an  expression  such  as  'the 
function  T  (£,  £)'  occurs  (cf.  42.03).  ...  I  call  the  places  in 
which  '£'  appears,  ^-argument-places,  and  those  in  which  '£' 
appears,  ^-argument- places.  I  say  that  the  ^-argument- 
places  are  mutually  cognate  and  similarly  the  ^-argument- 
places,  while  I  call  a  ^-argument-place  not  cognate  to  a  £- 

The  functions  with  two  arguments  \  =  £  and  \  >  £  always 
have  a  truth-value  as  value  [at  least  when  the  signs  '=' 
and  '>'  are  appropriately  explained].  For  our  purposes  we 
shall  call  such  functions  relations.  E.g.,  1  stands  to  1  in  the 
first  relation,  and  generally  every  object  to  itself,  while  2 
stands  to  1  in  the  second  relation.  We  say  that  the  object  T 
stands  to  the  object  A  in  the  relation  T  (£,  ?)  if  Y  (I\  A)  is  the 
True.  Similarly  we  say  that  the  object  A  falls  under  the  con- 
cept O  (£),  if  O  (A)  is  the  True.  It  is  naturally  presupposed 
here,  that  the  function  O  (£)  always  has  a  truth-value.  (Foot- 
note of  Frege' s :  A  difficulty  occurs  which  can  easily  obscure  the 
true  state  of  affairs  and  so  cast  doubt  on  the  correctness  of 
my  conception.  When  we  compare  the  expression  'the  truth- 
value  of  this,  that  A  falls  under  the  concept  O  (£)'  with  'O 
(A)',  we  see  that  to  '®  (  )'  there  properly  corresponds  'the 
truth-value  of  this,  that  (  )  falls  under  the  concept  O  (£)'  and 
not  'the  concept  O  (£)'.  Thus  the  last  words  do  not  properly 
signify  a  concept  [in  our  sense],  though  the  form  of  speech 
makes  it  seem  as  if  they  do.  As  to  the  difficulty  in  which 
language  thus  finds  itself,  cf.  my  paper  On  Concept  and  Object.) 

1.  Frege 

Variables,  introduced  by  Aristotle,  were  subsequently  regularly 
used  both  in  logic  and  mathematics.  A  reflective  concept  of  variable 
is  already  to  be  found  in  Alexander  of  Aphrodisias  (24.08V  In 
mathematical  logic  the  concept  of  variable  is  first  explicitly  intro- 
duced by  Frege. 



42.10  The  symbols  used  in  the  general  theory  of  magnitude 
fall  into  two  kinds.  The  first  consists  of  the  letters ;  each  letter 
represents  either  an  indeterminate  number  or  an  indeter- 
minate function.  This  indeterminateness  makes  it  possible  to 
express  by  means  of  letters  the  general  validity  of  propositions ; 
e.g.:  [a  +  b)  c  =  ac  +  be.  The  other  kind  contains  such 
symbols  as  +,  — ,  V,  0,  1,  2;  each  of  these  has  its  own  proper 

/  adopt  this  fundamental  idea  of  distinguishing  two  kinds  of 
symbols  (which  unfortunately  is  not  strictly  carried  out  in  the 
theory  of  magnitude  -  footnote  of  Frege's:  Consider  the 
symbols  1,  log,  sin,  Lim.  -)  in  order  to  make  it  generally 
applicable  in  the  wider  domain  of  pure  thought.  Accordingly, 
I  divide  all  the  symbols  I  use  into  those  that  can  be  taken  to 
mean  various  things  and  those  that  have  a  fully  determinate 
sense.  The  first  kind  are  letters,  and  their  main  task  is  to  be 
the  expression  of  generality.  For  all  their  indeterminateness, 
it  must  be  laid  down  that  a  letter  retains  in  a  given  context 
the  meaning  once  given  to  it. 

2.  Russell 

42.12  The  idea  of  a  variable,  as  it  occurs  in  the  present 
work,  is  more  general  than  that  which  is  explicitly  used  in 
ordinary  mathematics.  In  ordinary  mathematics,  a  variable 
generally  stands  for  an  undetermined  number  or  quantity.  In 
mathematical  logic,  any  symbol  whose  meaning  is  not  deter- 
minate is  called  a  variable,  and  the  various  determinations  of 
which  its  meaning  is  susceptible  are  called  the  values  of  the 
variable.  The  values  may  be  any  set  of  entities,  propositions, 
functions,  classes  or  relations,  according  to  circumstances. 
If  a  statement  is  made  about  'Mr  A  and  Mr  B\  'Mr  A'  and 
'Mr  BJ  are  variables  whose  values  are  confined  to  men.  A 
variable  may  either  have  a  conventionally-assigned  range  of 
values,  or  may  (in  the  absence  of  any  indication  of  the  range 
of  values)  have  as  the  range  of  its  values  all  determinations 
which  render  the  statement  in  which  it  occurs  significant.  Thus 
when  a  text-book  of  logic  asserts  that  lA  is  A',  without  any 
indication  as  to  what  A  may  be,  what  is  meant  is  that  any 
statement  of  the  form  'A  is  A'  is  true.  We  may  call  a  variable 
restricted  when  its  values  are  confined  to  some  only  of  those 
of  which  it  is  capable ;  otherwise  we  shall  call  it  unrestricted. 
Thus  when  an  unrestricted  variable  occurs,  it  represents  any 



object  such  that  the  statement  concerned  can  be  made 
significantly  (i.e.  either  truly  or  falsely)  concerning  that 
object.  For  the  purposes  of  logic,  the  unrestricted  variable  is 
more  convenient  than  the  restricted  variable,  and  we  shall 
always  employ  it.  We  shall  find  that  the  unrestricted  variable 
is  still  subject  to  limitations  imposed  by  the  manner  of  its 
occurrence,  i.e.  things  which  can  be  said  significantly  con- 
cerning a  proposition  cannot  be  said  significantly  concerning 
a  class  or  a  relation,  and  so  on.  But  the  limitations  to  which 
the  unrestricted  variable  is  subject  do  not  need  to  be  explic- 
itly indicated,  since  they  are  the  limits  of  significance  of  the 
statement  in  which  the  variable  occurs,  and  are  therefore 
intrinsically  determined  by  this  statement.  This  will  be  more 
fully  explained  later. 

To  sum  up,  the  three  salient  facts  connected  with  the  use 
of  the  variables  are:  (1)  that  a  variable  is  ambiguous  in  its 
denotation  and  accordingly  undefined;  (2)  that  a  variable 
preserves  a  recognizable  identity  in  various  occurrences 
throughout  the  same  context,  so  that  many  variables  can 
occur  together  in  the  same  context  each  with  its  separate 
identity;  and  (3)  that  either  the  range  of  possible  determina- 
tions of  two  variables  may  be  the  same,  so  that  a  possible 
determination  of  one  variable  is  also  a  possible  determination 
of  the  other,  or  the  ranges  of  two  variables  may  be  different, 
so  that,  if  a  possible  determination  of  one  variable  is  given 
to  the  other,  the  resulting  complete  phrase  is  meaningless 
instead  of  becoming  a  complete  unambiguous  proposition 
(true  or  false)  as  would  be  the  case  if  all  variables  in  it  had 
been  given  any  suitable  determinations. 


Truth-values,  which  are  of  great  importance  in  formal  logic, 
form  a  special  kind  of  value.  The  idea  is  already  present  in  the 
Megarian  school  (20.07),  but  its  expression  and  first  description 
comes  from  Frege.  His  doctrine  is  linked  to  his  own  semantics, 
according  to  which  every  proposition  is  a  name  for  truth  or  falsity, 
and  in  this  he  has  not  been  generally  followed,  but  the  concept  of 
truth-value  has  been  accepted  by  all. 

We  give  first  a  text  of  Frege's : 

42.13  But  that  indicates  at  the  same  time  that  the  domain 
of  values  for  functions  cannot  remain  limited  to  numbers ;  for  if 



I  take  as  arguments  of  the  function  £2  =  4  the  numbers  0,1,2,  3, 
in  succession,  I  do  not  get  numbers.  '02  =  4',  '12  =  4',  '22  =  4', 
'32  =  4'5  are  expressions  now  of  true,  now  of  false  thoughts. 
I  express  this  by  saying  that  the  value  of  the  function  £2  =  4 
is  the  truth-value  either  of  what  is  true  or  of  what  is  false. 
From  this  it  can  be  seen  that  I  do  not  intend  to  assert  any- 
thing by  merely  writing  down  an  equation,  but  that  I  only 
designate  a  truth-value;  just  as  I  do  not  intend  to  assert 
anything  by  simply  writing  down  '22'  but  only  designate 
a  number.  I  say:  The  names  '22  =4'  and  '3  >2'  stand  for  the 
same  truth-value'  which  I  call  for  short  the  True.  In  the  same 
manner  '32  =  4'  and  '1  >  2'  stand  for  the  same  truth-value, 
which  I  call  for  short  the  False,  just  as  the  name  '22'  stands  for 
the  number  4.  Accordingly  I  say  that  the  number  4  is  the 
reference  of  '4'  and  of  '22',  and  that  the  True  is  the  reference 
of  '3  >  2'.  But  I  distinguish  the  sense  of  a  name  from  its 
reference.  The  names  '22'  and  '  2+  2'  have  not  the  same  sense, 
nor  have  '22  =  4'  and  '2  +  2  =  4'.  The  sense  of  the  name  for  a 
truth-value  I  call  a  thought.  I  say  further  that  a  name  expresses 
is  its  sense,  and  what  it  stands  for  is  its  reference.  I  designate 
by  a  name  that  which  it  stands  for. 

The  function  £2  =  4  can  thus  have  only  two  values,  the 
True  for  the  arguments  +  2  and  -  2  and  the  False  for  all  other 

Also  the  domain  of  what  is  admitted  as  argument  must  be 
extended  -  indeed,  to  objects  quite  generally.  Objects  stand 
opposed  to  functions.  I  therefore  count  as  an  object  everything 
that  is  not  a  function:  thus,  examples  of  objects  are  numbers, 
truth-values,  and  the  ranges  to  be  introduced  further  on.  The 
names  of  objects  -  or  proper  names  -  are  not  therefore  accom- 
panied by  argument-places,  but  are  'saturated',  like  the 
objects  themselves. 

42.14  I  use  the  words,  'the  function  0(5)  has  the  same 
range  as  the  function  Y(5)',  to  stand  for  the  same  thing  as 
the  words,  'the  functions  0(5)  and  T(£)  have  the  same  value 
for  the  same  arguments'.  This  is  the  case  with  the  functions 
52  =  4  and  3.£2  =  12,  at  least  if  numbers  are  taken  as  argu- 
ments. But  we  can  further  imagine  the  signs  of  evolution  and 
multiplication  defined  in  such  a  manner  that  the  function 
(52  =  4)  =  (3.£2  =  12)  has  the  True  as  its  value  for  any 
argument  whatever.  Here  an  expression  of  logic  may  also  be 
used:  'The  concept  square  root  of  4  has  the  same  extension  as 



the  concept  something  whose  square  when  trebled  makes  12\  With 
those  functions  whose  value  is  always  a  truth-value  we  can 
therefore  say  'extension  of  the  concept'  instead  of  'range  of  the 
function',  and  it  seems  suitable  to  say  that  a  concept  is  a 
function  whose  value  is  always  a  truth-value. 

Independently  of  Frege,  Peirce  developed  similar  thoughts  in 
1885.  His  treatment  of  truth-values  is  more  formalistic  and  not 
tied  to  any  particular  semantic  theory.  However,  this  formalism 
enabled  him  to  formulate  one  which  seems  to  qualify  him  to  be 
regarded  as  a  precursor  of  many-valued  logics. 

42.15  According  to  ordinary  logic,  a  proposition  is  either 
true  or  false,  and  no  further  distinction  is  recognized.  This 
is  the  descriptive  conception,  as  the  geometers  say;  the  metric 
conception  would  be  that  every  proposition  is  more  or  less 
false,  and  that  the  question  is  one  of  amount.  At  present  we 
adopt  the  former  view. 

42.16  Let  propositions  be  represented  by  quantities.  Let 
v  and  f  be  two  constant  values,  and  let  the  value  of  the  quan- 
tity representing  a  proposition  be  v  if  the  proposition  is  true 
and  be  f  if  the  proposition  is  false.  Thus,  x  being  a  proposition, 
the  fact  that  x  is  either  true  or  false  is  written 

(x  -  f)  (v  -  x)  =  0. 

(x  -  f)  (v  -  g)  =  0 
will  mean  that  either  x  is  false  or  y  is  true.  .  .  . 

42.17  We  are,  thus,  already  in  possession  of  a  logical 
notation,  capable  of  working  syllogism.  Thus,  take  the 
premisses,  'if  x  is  true,  y  is  true',  and  'if  y  is  true,  z  is  true'. 
These  are  written 

(x  -  f)  (v  -  g)  =  0 
(y  -  f)  (v  -  z)   =  0. 
Multiply  the  first  by  (v  -  z)  and  the  second  by  [x  -  f)  and  add. 
We  get 

(x  -  I)  (v  -  f)  (v  -  z)  =  0, 
or  dividing  by  v  -  f,  which  cannot  be  0, 
(x  -  f)  (v  -  z)  =  0; 
and  this  states  the  syllogistic  conclusion,  'if  x  is  true,  z  is 

42.18  But  this  notation  shows  a  blemish  in  that  it  express- 
es propositions  in  two  distinct  ways,  in  the  form  of  quan- 
tities, and  in  the  form  of  equations;  and  the  quantities  are 



of  two  kinds,  namely  those  which  must  be  either  equal  to 
f  or  to  v,  and  those  which  are  equated  to  zero.  To  remedy 
this,  let  us  discard  the  use  of  equations,  and  perform  no 
operations  which  can  give  rise  to  any  values  other  than  f 
and  v. 

42.19  Of  operations  upon  a  simple  variable,  we  shall  need 
but  one.  For  there  are  but  two  things  that  can  be  said  about 
a  single  proposition,  by  itself;  that  it  is  true  and  that  it  is 

x  =   v  and  x  =  f. 
The  first  equation  is  expressed  by  x  itself,  the  second  by  any 
function,  9,  of  x,  fulfilling  the  conditions 
cpv  =  f     <pf  =  v. 
This  simplest  solution  of  these  equations  is 
yx  =  i  +  v  -  x. 

1.  Peirce 

The  standpoint  revealed  in  the  last  text  comes  near  to  defining 
propositional  functors  by  means  of  truth-values.  Tabular  definitions 
of  this  kind  have  already  been  met  with  in  the  Stoic-Megarian  school 
(20.07 ff.),  later  on  in  Boole,  though  without  explicit  reference  to 
truth-values,  and  finally  in  Frege's  Begriffsschrift  (41.12).  Peirce  has 
the  notion  quite  explicitly,  and  in  connection,  moreover,  with 
ancient  logic,  in  1880: 

42.20  There  is  a  small  theorem  about  multitude  that  it 
will  be  convenient  to  have  stated,  and  the  reader  will  do  well 
to  fix  it  in  his  memory  correctly.  ...  If  each  of  a  set  of  m 
objects  be  connected  with  some  one  of  a  set  of  n  objects,  the 
possible  modes  of  connection  of  the  sets  will  number  nm. 
Now  an  assertion  concerning  the  value  of  a  quantity  either 
admits  as  possible  or  else  excludes  each  of  the  values  v  and  f. 
Thus,  v  and  f  form  the  set  m  objects  each  connected  with  one 
only  of  n  objects,  admission  and  exclusion.  Hence  there  are, 
nm ,  or  22,  or  4,  different  possible  assertions  concerning  the 
value  of  any  quantity,  x.  Namely,  one  assertion  will  simply 
be  a  form  of  assertion  without  meaning,  since  it  admits  either 
value.  It  is  represented  by  the  letter,  x.  Another  assertion 
will  violate  the  hypothesis  of  dichotomies  by  excluding  both 
values.  It  may  be  represented  by  x.  Of  the  remaining  two, 
one  will  admit  v  and  exclude  f,  namely  x;  the  other  will 
admit  f  and  exclude  v,  namely  x. 



Now,  let  us  consider  assertions  conce- 
rning the  values  of  two  quantities,  xand 
y.  Here  there  are  two  quantities,  each 
of  which  has  one  only  of  two  values;  so 
that  there  are  22,  or  4,  possible  states  of 
things,  as  shown  in  this  diagram. 

Above  the  line,  slanting  upwards  to 
the  right,  are  placed  the  cases  in  which 
x  is  v;  below  it,  those  in  which  x  is  f. 
Above  the  line  but  slanting  downward 

to  the  right,  are  placed  the  cases  in  which  y  is  v;  below  it, 
those  in  which  y  is  f.  Now  in  each  possible  assertion  each  of 
these  states  of  things  is  either  admitted  or  excluded ;  but  not 
both.  Thus  m  will  be  22,  while  n  will  be  2;  and  there  will 
be  nm,  or  22,  or  16,  possible  assertions.  .  .  . 

Of  three  quantities,  there  are  23,  or  8,  possible  sets  of 
values,  and  consequently  28,  or  256,  different  forms  of  propo- 
sitions. Of  these,  there  are  only  38  which  can  fairly  be  said  to 
be  expressible  by  the  signs  [used  in  a  logic  of  two  quantities]. 
It  is  true  that  a  majority  of  the  others  might  be  expressed 
by  two  or  more  propositions.  But  we  have  not,  as  yet,  expressly 
adopted  any  sign  for  the  operation  of  compounding  propo- 
sitions. Besides,  a  good  many  propositions  concerning  three 
quantities  cannot  be  expressed  even  so.  Such,  for  example, 
is  the  statement  which  admits  the  following  sets  of  values : 

x     y     z 


V       V 


f    f 


V       f 


f        V 

Moreover,  if  we  were  to  introduce  signs  for  expressing 
[each  of]  these,  of  which  we  should  need  8,  even  allowing  the 
composition  of  assertions,  still  16  more  would  be  needed  to 
express  all  propositions  concerning  4  quantities,  32  for  5, 
and  so  on,  ad  infinitum. 

2.  Wittgenstein 

The  same  doctrine  was  systematically  elaborated  about  1920  by 
J.  Lukasiewicz,  E.  L.  Post  and  L.  Wittgenstein.  We  give  the  relevant 
text  from  the  last: 

42.21.  With  regard  to  the  existence  of  n  atomic  facts  there 




Kn  =  2   (    )  possibilities. 

It  is  possible  for  all  combinations  of  atomic  facts  to  exist, 
and  the  others  not  to  exist. 

42.22  To  these  combinations  correspond  the  same  number 
of  possibilities  of  the  truth  -  and  falsehood  -  of  n  elementary 

42.23  The  truth-possibilities  of  the  elementary  propositions 
mean  the  possibilities  of  the  existence  and  non-existence  of 
the  atomic  facts. 

42.24  The  truth-possibilities  can  be  presented  by  schemata 
of  the  following  kind  ('T'  means  'true,  lF'  'false'.  The  rows 
of  T's  and  F's  under  the  row  of  the  elementary  propositions 
mean  their  truth-possibilities  in  an  easily  intelligible  sym- 

P  q  r 

T  T  T 

F  T  T 

~T~  F  _T 

T  T  F 

F~  F  ~f 

F T  F 

T  F  F 

F  F  F 

p <L 

T T 

F T 

T       F 
F       F 

42.25  .  .  .  The  truth-possibilities  of  the  elementary  propo- 
sitions are  the  conditions  of  the  truth  and  falsehood  of  the 

42.26  It  seems  probable  even  at  first  sight  that  the  intro- 
duction of  the  elementary  propositions  is  fundamental  for 
the  comprehension  of  the  other  kinds  of  propositions.  Indeed 
the  comprehension  of  the  general  propositions  depends  palpably 
on  that  of  the  elementary  propositions. 

42.27  With  regard  to  the  agreement  and  disagreement  of  a 
proposition    with    the    truth-possibilites    of    n    elementary 

Kn    (K  \ 
propositions  there  are    2    (    n)  =   Ln  possibilities. 

42.28  .  .  .  Thus  e.g. 
















is  a  prepositional  sign. 

42.29  .  .  .  Among  the  possible  groups  of  truth-conditions 
there  are  two  extreme  cases. 

In  the  one  case  the  proposition  is  true  for  all  the  truth- 
possibilities  of  the  elementary  propositions.  We  say  that  the 
truth-conditions  are  tautological. 

In  the  second  case  the  proposition  is  false  for  all  the  truth- 
possibilities.  The  truth-conditions  are  self-contradictory. 

In  the  first  case  we  call  the  proposition  a  tautology,  in  the 
second  case  a  contradiction. 

42.30  The  proposition  shows  what  it  says,  the  tautology 
and  the  contradiction  that  they  say  nothing. 

The  tautology  has  no  truth-conditions,  for  it  is  uncondi- 
tionally true;  and  the  contradiction  is  on  no  condition  true. 

Tautology  and  contradiction  are  without  sense. 

(Like  the  point  from  which  two  arrows  go  out  in  opposite 

(I  know,  e.g.  nothing  about  the  weather,  when  I  know 
that  it  rains  or  does  not  rain.) 

42.31  Tautology  and  contradiction  are,  however,  not 
nonsensical;  they  are  part  of  the  symbolism,  in  the  same 
way  that  '0'  is  part  of  the  symbolism  of  Arithmetic. 

The  name  'tautology'  and  the  last  quotation  show  the  peculiar 
(extremely  nominalist)  tendency  underlying  Wittgenstein's  semantic 
views.  It  is  diametrically  opposed  to  Frege's  tendency  and  from  his 
point  of  view  misleading. 


The  tables  of  values  constructed  in  the  texts  just  cited  provide  a 
decision  procedure  for  propositional  functions,  i.e.  a  procedure  which 
enables  one  to  decide  whether  a  function  is  a  logical  law  (whether  it 
becomes  a  true  proposition  for  every  correct  substitution).  The  basic 
idea  of  such  a  procedure  is  present  in  Schroder  (42.32).  It  was 
developed  by  E.L.  Post  (42.33)  and  was  known  to  J.  Lukasiewicz  at 
the  same  time  (42.34).  It  is  set  out  in  full  in  the  manuals  of  Hilbert 



and  Ackermann  (42.35),  1928,  and  T.  Kotarbinski,  1929.  We  quote 
Kotarbinski's  text,  because  of  its  clarity.  The  author  writes  'p"  for 
'not  p',  and  uses  '+',  '<',  '  =  ',  as  signs  of  addition,  implication  and 
equivalence  respectively. 

42.36  We  shall  now  give  a  very  simple  method  of  verifica- 
tion for  the  propositional  calculus,  which  enables  one  to 
verify  the  correctness  of  every  formula  in  this  domain  [viz. 
the  zero-one  method  of  verification].  We  stipulate  for  this 
purpose  that  it  is  permitted  to  write,  say,  zero  for  a  false 
proposition,  and  one  for  a  true.  With  the  help  of  this  sym- 
bolism we  now  investigate  whether  a  given  formula  becomes 
a  true  proposition  for  all  substitutions  of  propositions  for 
propositional  variables  -  always  under  the  condition  that  the 
same  (proposition)  is  substituted  for  the  same  (variable),  or 
whether  on  the  other  hand  it  becomes  a  false  proposition  for 
some  substitutions.  In  the  first  case  it  is  a  valid  formula,  in 
the  second  an  invalid  one.  .  .  . 

We  recollect  in  this  connection:  (1)  that  the  negation  of  a 
true  proposition  is  always  a  false  proposition,  and  conver- 
sely; (2)  that  the  logical  product  is  true  only  when  both 
factors  are  true;  (3)  that  the  logical  sum  is  always  true  when 
at  least  one  of  its  parts  is  true,  and  false  only  in  the  case 
that  both  its  parts  are  false;  (4)  that  implication  is  false 
only  when  its  antecedent  is  true  and  its  consequent  false; 
(5)  that  equivalence  is  true  only  when  both  sides  are  true  or 
both  false;  but  when  one  side  is  true,  the  other  false,  then  the 
whole  equivalence  is  false.  If,  for  example,  we  put  zero  for  p 
and  one  for  q  in  a  formula,  then  this  formula  can  be  further 
simplified  by  writing  a  zero  in  place  of  the  product  of  p  and  q 
wherever  this  occurs  throughout  the  formula-  and  analogously 
in  the  case  of  the  other  functions.  .  .  . 

If,  after  the  application  of  all  possible  substitutions  of 
zero  and  one  for  propositional  variables  in  a  given  formula, 
and  after  carrying  out  the  .  .  .  simplifications  described,  the 
formula  always  reduces  to  a  one,  then  it  is  true.  But  if  it 
is  zero  for  even  a  single  choice  of  substitutions,  it  is  invalid. 

To  have  an  example,  we  verify  the  formula  of  transpo- 
sition .  .  . 

(p  <  q)  =  W  <  p') 

1.  We  suppose  that  true  propositions  have  been  substituted 
both  for  p  and  for  q.  Our  formula  then  takes  on  the  form 

(i  <  i)  =  (r  <  i'). 



Simplifying  it  by  Rule  (1),  we  get: 

(1<1)  =  (0<0) 
.  .  .  further  ...  we  get 

1  =  1 
which  ...  we  can  replace  by 


The    process   is   then    repeated    with    the   other   three    possible 


In  the  last  two  paragraphs  we  have  spoken  of  the  basic  concepts 
and  one  of  the  main  methods  of  modern  propositional  logic.  Now  we 
come  on  to  show  the  second,  axiomatic,  method  at  work,  in  some 
sample  sections  reproduced  from  different  propositional  systems. 

These  sections  will  be  taken  from  the  systems  of  McColl  (1877), 
Frege  (1879  and  1893),  Whitehead  and  Russell  (1910)  -  here  we  insert 
a  text  from  Peirce,  connected  with  the  Sheffer  stroke  -  and  finally 
Lukasiewicz  (1920).  In  this  last  system,  two-valued  propositional 
logic  seems  to  have  reached  the  term  of  its  development. 


The  ensuing  texts,  which  are  a  continuation  of  the  definitions  in 
41.05 ff.,  contain  rules  for  an  algebraic  system  of  propositional  logic, 
constructed  in  the  spirit  of  Boole.  It  may  be  compared  with  the 
non-algebraic  system  of  Lukasiewicz  (43.45).  It  may  be  remarked 
that  while  this  algebraic  style  has  been  for  the  most  part  superseded, 
there  have  also  been  quite  recent  algebraic  systems  (Tarski). 

43.01  Rule  1.  -  The  rule  of  ordinary  algebraical  multi- 
plication applies  to  the  multiplication  of  indeterminate 
statements,  thus : 

A{B  +  C)  =AB  +  AC;  (A  +  B)  (C  +  D)  =  AC  +  AD  +  BC  + 


and  so  on  for  any  number  of  factors,  and  whatever  be  the 

number  of  terms  in  the  respective  factors. 

43.02  Rule  2.  -  Let  A  be  any  statement  whatever,  and  let 
B  be  any  statement  which  is  implied  in  A  [and  which  must 
therefore  be  true  when  A  is  true,  and  false  when  A  is  false]; 
or  else  let  B  be  any  statement  which  is  admitted  to  be  true 
independently  of  A;  then  [in  either  case]  we  have  the  equation 
A  =  AB.  As  particular  cases  of  this  we  have  A  =  AA  = 



AAA   =  etc.,  as  repetition  neither  strengthens  nor  weakens 
the  logical  value  of  a  statement.  Also, 
A  =A{B+B')=A{B+B'){C+C')=etc.,  for 
B+B'  =  l=C+C'=etc.  [see  Def.  4]  (41.08). 

43.03  Rule  3.  -  {ABy=AB,+AfB+A,B' 

=A'  B+B'(A+A')=A'  B+B' , 
for  A+A'  —  l  and  B  +  B'  =  1.  Similarly  we  may  obtain  various 
equivalents  (with  mutually  inconsistent  terms)   for  (ABC)', 
(ABCD)',  etc. 

43.04  Rule  4.  -  [A+R)'  =  A' B' ;  {A+B+C)'  =  A'B'C ;  and 
so  on. 

43.05  Rule  5. -A+B     ={(A+B)'}' =  {A'B')' 

=A'B+A(B'+B  =  A'B+A. 

Similarly   we    get    equivalents    (with    mutually   inconsistent 

terms)  for  A+B+C,A+B+C+D,  etc. 

43.06  Rule  11.  -  If  A:B,  then  B':A'.  Thus  the  implica- 
tions A:B  and  B':A'  are  equivalent,  each  following  as  a 
necessary  consequence  of  the  other.  This  is  the  logical  prin- 
ciple of  'contraposition'. 

43.07  Rule  12.  -  If  A:B,  then  AC:BC,  whatever  the  state- 
ment C  may  be. 

43.08  Rule  13.  -  If  A:*,  B:$,  C:y,  then  ABC: a^y,  and  so 
on  for  any  number  of  implications. 

43.09  Rule  14.  -  If  AB=0,  then  A:B'  and  B:A' . 

43.10  Definition  13.  -  The  symbol  A+B  asserts  that  A  does 
not  imply  B;  it  is  thus  equivalent  to  the  less  convenient 
symbol  (A: By. 

43.11  Rule  15.  -  If  A  implies  B  and  B  implies  C,  then  A 
implies  C. 

43A2  Bute  16.  -  If  A  does  not  imply  B7  then  B'  does  not 
imply  A';  in  other  words,  the  non-implications  A+B  and 
B'+Af  are  equivalent. 

43.13  Rule  17.  -  If  A  implies  R  but  does  not  imply  C,  then 
B  does  not  imply  C  in  other  words,  from  the  two  premises 
A:B  and  A+C,  we  get  the  conclusion  B+C. 

43.14  The  following  formulae  are  all  either  self-evident 
or  easily  verified,  and  some  of  them  will  be  found  useful  in 
abbreviating  the  operations  of  the  calculus :  - 

(1)  i<  =  0,0'  =  l; 

(2)  l  =  l+a  =  l+a  +  b  =  l+a  +  b  +  c,  etc. ; 



(3)  {ab  +  a'b')'  =  a'b  +  ab' 

(a'b  +  ab')'  =  ab  +  a'b' 




a  :  a  +  b  :  a  +  b  +  c,  etc. ; 

(a  +  A)  (a+  D)  (a  +  C)  ...  =  a  +  A£C  ...; 

(a  :  b)  :  a'  +  b; 

(a  =  b)  =  (a  ib)  {b  :  a); 

(a  —  b)  :  ab  +  a'b' ; 

(A  :a)  (5:6)  (C  :  c)  ...  :  (ABC  ...  :  afo  ...); 

(A  :  a)  (  £  :  b)  (  C:  c)  ... :  [A  +  £  +  C  ...:a  +  b  +  c  + 


(A  :aj)  [B:x)  {C  :  x)  ...  =  (A  +  B  +  C  +  ...  :x); 

(x:A)  {x:  B)  (as   C)  ...  =  (x:i5C  ...); 

(A  :aj)  +  (B:«)  +  (C  :  x)  +  ...  :  {ABC  ...  :a?); 

(x  :  A)  +  [x  :  B)  +  (x  :  C)  +  ...  :  (x  :  A  +  B  +  C  +  ...). 


One  of  Frege's  most  important  intuitions  concerned  the  distinc- 
tion between  theorems  and  rules  of  inference.  This  is  already  to  be 
found  in  the  Begriffsschrift  (43.15),  where  he  uses  only  a  single 
rule;  in  the  Grundgesetze  he  adopts  several  for  reasons  of  practical 
convenience,  of  which  we  give  four: 

43.16  //  the  lower  member  of  a  proposition  differs  from  a 
second  proposition  only  in  lacking  the  judgment-stroke,  one 
can  conclude  to  another  proposition  which  results  from  the  first 
by  suppression  of  that  lower  member. 

'Lower  member',  i.e.  antecedent.  The  sense  is  therefore:  Given 
'  | — if  B,  then  A'  and  further  '  | — B\  then  we  may  suppress  B  in 
the  conditional  proposition  to  obtain  '  | — A\  This  is  the  modus 
ponendo  ponens  (22.04). 

43.17  A  lower  member  may  be  exchanged  with  its  upper 
member,  if  at  the  same  time  the  truth-value  of  each  is  changed. 

Thus,  given  'if  B,  then  A\  one  may  write  'if  not  A,  then  not  B' ; 
this  is  the  rule  of  simple  contraposition  (31.17,  cf.  43.22  [28]). 

43.18  //  the  same  combination  of  symbols  occurs  as  upper 
member  in  one  proposition  and  lower  member  in  another,  one 
can  conclude  to  a  proposition  in  which  the  upper  member  of 
the  second  appears  as  upper  member,  and  all  lower  members  of 
the  two,  save  the  one  mentioned,  as  lower  members.  Bui  lower 
members  which  occur  in  both,  need  only  be  written  once. 



Given  'if  C,  then  B'  and  'if  B,  then  A\  one  may  write  'if  C,  then 
A\  This  is  the  rule  corresponding  to  the  law  of  syllogism  (cf.  31.18). 

43.19  //  two  propositions  correspond  in  their  upper  members, 
while  a  lower  member  of  one  differs  from  a  lower  member  of  the 
other  only  in  respect  of  a  preceding  negation-stroke,  then  we 
can  conclude  to  a  proposition  in  which  the  corresponding  upper- 
member  appears  as  upper  member,  and  all  lower  members  of  the 
two,  with  the  exception  of  the  two  mentioned,  as  lower  members. 
Lower  members  which  occur  in  both  are  only  to  be  written  down 

Frege's  concrete  example  (43.20)  is  this:  Given  'if  e,  then  if  not  d, 
then:  if  b,  then  a'  and  'if  e,  then  if  d,  then:  if  b,  then  a',  one  may 
write:  'if  e,  then:  if  b,  then  a'. 

Lukasiewicz,  deriving  from  Frege,  formulates  the  difference 
between  thesis  and  rule,  and  states  the  most  important  rule,  as 
follows : 

43.21  A  logical  thesis  is  a  proposition  in  which  besides 
logical  constants  there  occur  only  propositional  or  name- 
variables  and  which  is  true  for  all  values  of  the  variables  that 
occur  in  it.  A  rule  of  inference  is  a  direction  which  empowers 
the  maker  of  inference  to  derive  new  theses  on  the  basis  of 
already  admitted  theses.  Thus  e.g.  the  laws  of  identity  given 
above  are  logical  theses,  while  the  following  'rule  of  detach- 
ment' is  a  rule  of  inference : 

Whose  admits  as  true  the  implication  'if  a,  then  p'  and  the 
antecedent  'a'  of  this  implication,  has  the  right  to  admit  as 
true  also  the  consequent  '[}'  of  this  implication. 

Thus  for  Lukasiewicz,  'logical  thesis'  covers  both  axioms  and 
derived  propositions. 


Space  prevents  us  from  giving  Frege's  propositional  schemata 
(corresponding  to  Lukasiewicz's  theses)  in  the  original  symbolism; 
Instead,  we  translate  some  of  them  into  Peano-Russellian. 

43.22  01.  ud  .bD  a 

02.  cD.bDaiDicDb.D.cDa 

03.  bDa.Di.cD.bDaiDicDb.D.cDa 

04.  b  d  a  .  d  :  c  d  .  b  d  a  :  .  d  :  .  b  d  a  :  d  :  c  d  b  .  d  .  c 
d  a 

05.  bDaiDicDb.D.cDa 





a  re 
a  ~r 

a  ^w 
a  -    c~/> 

An  example  of  Frege's  Symbolism  taken  from  "Begriffsschrift",  p.  56. 


06.  c  d  .  b  d  a  :  .  d  :  .  c  :  d  :  d  d  b  .  d  .  d  d  a 

07.  b  d  a  :  .  d  :  .  d  d  .  c  d  b  :  d  :  d  .  d  .  c  d  a 

08.  d  d  .  b  d  a  :  d  :  b  .  d  .  d  d  a 

09.  c  d  b  :  d  :  b  d  a  .  d  .  c  d  a 

10.  e  d  .  d  d  b  :  d  a  :  .  d  .  d  d  .  e  d  b  :  d  a 

11.  CDb.D.aiD.bDCl 

12.  d  3  :  c  .  d  .  6  d  a  :  .  d  :  .  d  :  d  :  b  . 

13.  d  d  :  c  .  d  .  b  d  a  :  .  d  :  .  b  :  d  :  d  . 

d  .  d  .  c  d  a 

15.  m.diDic.DiDo:::::!*:.:: 
d  .  c  d  a 

d  .  c  d  a 
d  .  c  d  a 
:  .  D  :  .  6  :  d 



e  d  :  .  d  d  :  c  d  .b  d  a:  :  d  : :  e  d  :  .  d  d  :  b  d 








e  d  :  d  d  .  c  d  b  :  :  d  :  :  b  d  a:  .  d  :  .e  d  :  d  .  : 




f  d  :  :  eD  :  .  d  d  :  c  d  .b  d  a:: .  d  ::  .  f  :>  :  :  < 

d  :  b  d  .  c  d  a 


d  d  :  c  d  .  b  d  a  :  .  d  :  :  e  d  d .  d  :  .  c  d  :  b  .  z 






b  d  .  a  d  a 


a  d  a 








~  ~  a  d  a 











a  d  .  <— '  a  d  b 




~  a  .  d  .  a  d  b 






a  d  ~  ~  a 


~  ~  (a  d  a) 


~  a  d  a  .  d  .  a 

c  d  a 

c  d  a 

e  d  :  .  d 

e  d  a 



44.  ~aDc.D:cDa.Da 

45.  ~cDa.D.~aDc\D'.  .~cDa:D:cDa.Da 

46.  ~cDa.D:cDa.D.a 

47.  ~cDb.D:.bDa.D:cDa.Da 

48.  d  d  .  ~  c  d  b  :  d  :  .  b  d  a  .  d  :  c  d  a  .  d  .  d  d  a 

49.  ~cDb.D\.cDa.D:bDa.Da 

50.  CDa.D:.ha.D:~CDfi.Dfl 

51.  d  d  .  c  d  a:  d  ::b  d  a  .  d  :  .  d  d  :  ~  c  d  b  .  d  a 

52.  C   ^d.D.f{c)Df{d) 

53.  /  (c)  d  :  c  =  d  .  d  f  (d) 

54.  c  =  c 

55.  c  =  d  .  d  .  d  =  c 

56.  d=c.D.f(d)Df{c) 

57.  c  =d.D  .f{d)Df{c) 


Passing  over  Peano,  we  come  now  to  the  Principia  Malhematica 
of  Whitehead  and  Russell  (Vol.  1,  1910). 

1.  Primitive  symbols  and  definition 

Besides  variables,  Frege's  sign  of  assertion  V  and  Peano's  dots 
and  brackets,  the  Principia  uses  only  two  undefined  primitive 
symbols:  '~'  and  'v'.  'p'  is  read  as  'not  p',  'p  v  q'  as  'p  or  q'  the 
alternation  being  non-exclusive  (43.23). 

Implication  is  defined : 

43.24  *1.01.  p  d  q  .  =  .  ~  p  v  q  Di. 

2.  Axioms  (Primitive  Propositions) 

43.25  *1.1.  Anything  implied  by  a  true  elementary  propo- 
sition is  true.  Pp.  (Footnote  :  The  letters  "Pp"  stand  for  "primi- 
tive proposition",  as  with  Peano.) 

The  above  principle  ...  is  not  the  same  as  "if  p  is  true,  then  if 
p  implies  q,  q  is  true".  This  is  a  true  proposition,  but  it  holds 
equally  when  p  is  not  true  and  when  p  does  not  imply  q. 
It  does  not,  like  the  principle  we  are  concerned  with,  enable 
us  to  assert  q  simply,  without  any  hypothesis.  We  cannot 
express  the  principle  symbolically,  partly  because  any 
symbolism  in  which  p  is  variable  only  gives  the  hypothesis  that 
p  is  true,  not  the  fact  that  it  is  true. 

43.26  *1.2.  h  :pv  p.  Dp  Pp. 

This  proposition  states :  "If  either  p  is  true  or  p  is  true,  then 



p  is  true".  It  is  called  the  "principle  of  tautology",  and  will  be 
quoted  by  the  abbreviated  title  of  "Taut".  It  is  convenient 
for  purposes  of  reference,  to  give  names  to  a  few  of  the  more 
important  propositions;  in  general,  propositions  will  be  refer- 
red to  by  their  numbers. 

43.27  *1.3.  h  :q.  d  .  p  v  q  Pp. 

This  principle  states:  "If  q  is  true,  then  'p  or  7'  is  true". 
Thus  e.g.  if  q  is  "to-day  is  Wednesday"  and  p  is  "to-day  is 
Tuesday",  the  principle  states:  "If  to-day  is  Wednesday,  then 
to-day  is  either  Tuesday  or  Wednesday".  It  is  called  the 
"principle  of  addition".  .  .  . 

43.28  *1.4.  [-  :  p  v  q  .  d  .  q  v  p  Pp. 

This  principle  states  that  "p  or  q"  implies  llq  or  p".  It  states 
the  permutative  law  for  logical  addition  of  propositions,  and 
will  be  called  the  "principle  of  permutation".  .  .  . 

43.29  *1.5.  npv(gvr).3.?v(pvr)Pp. 

This  principle  states:  "If  either  p  is  true,  or  'q  or  r'  is  true, 
then  either  q  is  true,  or  'p  or  r'  is  true".  It  is  a  form  of  the 
associative  law  for  logical  addition,  and  will  be  called  the 
'associative  principle'.  .  .  . 

43.30  *1.6.  \-  ::  .  q  d  r  .  d  :  p  v  q  .  d  .  p  v  r  Pp. 

This  principle  states:  "If  q  implies  r,  then  'p  or  q'  implies 
*p  or  p'  ".  In  other  words,  in  an  implication,  an  alternative 
may  be  added  to  both  premiss  and  conclusion  without 
impairing  the  truth  of  the  implication.  The  principle  will  be 
called  the  "principle  of  summation",  and  will  be  referred  to 
as  "Sum". 

3.  Statement  of  proofs 

Two  examples  will  explain  the  method  of  proof  used  in  the 

43.31  *2.02.  1-  :q.D  .  p  Dq 


< — '  p 
Add \-  :  q  .  d  .  ~  pv    q  (1) 

(1)  .(*1.01)  h  :q.  D.p  Dq 

This  is  to  be  read:  take  'Add',  i.e.  43.27: 

q.  D.p  v  q 

and  in  it  substitute  '~  p'  for  'p';  we  obtain 



As  according  to  43.24  '~  p  v  q'  and  'p  d  q'  have  the  same  meaning, 
the  latter  can  replace  the  former  in  ( 1 ) ,  which  gives  the  proposition  to 
be  proved;  it  corresponds  to  the  Scholastic  verum  sequitur  ad  quod- 

That  proof  is  a  very  simple  one;  a  slightly  more  complicated 
example  is: 

43.32   *2.3.  h  :p  v  (q  v  r) .  d  .  p  v  (r  v  q) 


q  v  r,  r  v  q 

h  :  q  v  r  •  d  .  r  v  q: 

d  t-  :  p  v  (q  v  r)  .  d  .p  v  (r  v  q) 

4.  Laws 



43.33  The    most    important    propositions    proved 
present  number  are  the  following :  .  .  . 
*2.03.   \-:pD~q.D.qD~p 
*2.15.   \-:~pDq.D.~qDp 
*2.16.   \-:pDq.D.~qD~p 
*2.17.  h:~p^p.:.p:^ 

These    four    analogous    propositions    consitute  the   "prin- 
ciple of  transposition"  .  .  . 
*2.04.   h  :.  p  .  d  .  q  d  r  :  d  :  q  .  d  .  p  d  r 
*2.05.   h  i.qDr.DipDq.D.pDr 
*2.06.   h  i.pDq.DiqDr.D.pDr 

These  two  propositions  are  the  source  of  the  syllogism  in 
Barbara  (as  will  be  shown  later)  and  are  therefore  called  "the 
principle  of  the  syllogism"  .  .  . 
*2.08.  \-  .  pD  p 

I.e.  any  proposition  implies  itself.  This  is  called  the  "prin- 
ciple of  identity"  .  .  . 
*2.21.   \-  :  ~  p  .  d  .  p  d  q 

I.e.  a  false  proposition  implies  any  proposition. 

Next  the  Principia  gives  a  series  of  laws  concerning  the  logical 
product  (43.34).  At  their  head  stand  the  two  definitions: 

43.35  *3.01.  p  .  q  .  =  .  —  [~  p  v  —  q)  Df 
where  "p  .  q"  is  the  logical  product  of  p  and  q. 
*3.02.  pDqDr.  =  .pDq.qDrDf. 

This  definition  serves  merely  to  abbreviate  proofs. 

43.36  The  principal  propositions  of  the  present  number  are 
the  following: 



*3.2.  h  :.  p  ,  d  :  q  .  d  .  p  .  q 

I.  e.  "p  implies  that  q  implies  p  .  ry",  i.e.  if  each  of  two  pro- 
positions is  true,  so  is  their  logical  product. 
*3.26.   \-  :  p  .q  .  d  .  p 
*3.27.  h  :  p  .  q .  p  .  q 

I.e.  if  the  logical  product  of  two  propositions  is  true,  then 
each  of  the  two  propositions  severally  is  true. 
*3.3.   h  :.  p  .  q  .  d  .  r  :  d  :  p  .  d  .  q  d  r 

I.e.  if  p  and  q  jointly  imply  r,  then  p  implies  that  q  implies 
r.  This  principle  (following  Peano)  will  be  called  "exportation", 
because  q  is  "exported"  from  the  hypothesis  .  .  . 
*3.31.   h  :.  p  .  d  .  q  d  r  :  d  :  p  .  q  .  d  .  r  .  .  . 
*3.35.   I-  :  p  .